Kerem Ege
email: kerem.ege@polytechnique.edu

Xavier Boutillon

Synthetic description of the piano soundboard mechanical mobility

INTRODUCTION

The bridge of a piano soundboard is the location where the energy of the strings is transferred to the soundboard. This transfer is ruled by the end condition of the strings which is described here by a 1-D mechanical mobility or admittance Y (ω) = V (ω)/F(ω). The purpose of this communication is to give an expression of the piano soundboard mechanical mobility depending on a small number of parameters and valid up to several kHz. In the first section we describe the coupling, and a bibliographical review of the published measurements is given. The synthetic description derived from Skudrzyk's and Langley's work is then introduced and applied to an upright piano: the average driving point mobility and its envelope are expressed with only the modal density n( f ), the mean loss factor η( f ) and the mass M of the structure. The measurements of the modal density and the modal loss factors were done up to 2.5 kHz, with a novel high-resolution modal analysis technique presented in our ICA companion-paper Vibrational and acoustical characteristics of the piano soundboard devoted to the vibration and some radiation characteristics of the soundboard [START_REF] Ege | Vibrational and acoustical characteristics of the piano soundboard[END_REF]. Finally, we present in the last section how it can be used to predict the changes of the driving point mobility, and possibly of the sound radiation in the treble range, resulting from structural modifications of the piano soundboard.

MECHANICAL MOBILITY Coupling

The research of a trade-off between loudness and sustain (duration) is a major issue for piano designers and manufacturers. The way by which the energy of vibration is transferred from the piano string to the soundboard depends in particular on the end conditions of the strings at the bridge. This is a classical problem of impedance matching between a source (the string) and a load (the soundboard).

The mechanical load presented to the string can be described by the admittance Y (ω) (also called mechanical mobility) at the connecting point (bridge) between the string and the soundboard. The admittance defines the relationship between the local velocity V and the excitation force F. Since these quantities are both of vectorial nature, Y is a 3 × 3 matrix. In principle, the reciprocal quantity -the impedance Z = Y -1 could be used as well. In most mechanical systems however (including musical string instruments) the force is imposed in one direction only and the other directions are left totally free. These cases are described by only three coefficients of the mobility matrix (see for example [START_REF] Boutillon | Three-dimensional mechanical admittance: Theory and new measurement method applied to the violin bridge[END_REF]).If, in addition, only one direction of motion is under investigation, only one mobility coefficient needs to be known. In what follows, the notation Y means the ratio between the velocity and the force in the direction normal to the soundboard. It would be the Y zz coefficient of the full matrix and one should notice in the line of the previous discussion that Z zz = Y -1 zz .

The characteristic mobility of transverse waves in a piano string is always much higher than that of the soundboard. Literature more often deals with the characteristic impedance and with the impedance-like quantity Y -1 at the bridge. The former ranges typically from ≈ 10 kg s -1 for the long and thick bass strings to ≈ 5 kg s -1 in the treble range [START_REF] Askenfelt | Sound radiation and timbre[END_REF]). The latter is almost 100 times larger with an average low-frequency level near 10 3 kg s -1 . According to Askenfelt [START_REF] Askenfelt | Sound radiation and timbre[END_REF], piano makers have reached this value empirically since [it] gives the proper amount of coupling and decay rate suitable for musical purposes. Pianos having a larger soundboard mobility at the bridge tend to sound harsh and to exhibit less than normal durations of tones according to Conklin [START_REF] Conklin | Design and tone in the mechanoacoustic piano. Part 2. Piano structure[END_REF]. Conversely, if the mobility level falls significantly lower, the duration is longer than normal while, at the same time, the output seems subnormal. Wogram published the first impedance measurements [START_REF] Wogram | Acoustical Research on Pianos. Part I: Vibrational Characteristics of the Soundboard[END_REF]. He used an electrodynamic shaker to drive the board and an impedance head to measure at the same point the excitation force and vibration velocity. Typical results near the centre of the board are reported in figure 1. The resonances in the soundboard motion appear as the minima of the impedance magnitude, corresponding to ϕ = 0 • in the -π/2 → π/2 phase transition. Between 100 and 1000 Hz, the average value of the impedance is roughly 10 3 kg s -1 . Above this range, |Z| decreases uniformly at a rate of about 5 dB per octave to approximately 160 kg s -1 at 10 4 Hz. This rapid falloff, almost inversely proportional to frequency, appears as a measurement artefact: it has the definite appearance of some purely springy impedance which is somehow appearing in parallel with the measured one, according to Weinreich [START_REF] Weinreich | Vibration and radiation of structures with application to string and percussion instrument[END_REF]. Giordano [START_REF] Giordano | Mechanical impedance of a piano soundboard[END_REF] adds: it could have been caused by an effective decoupling of his impedance head from the soundboard at high frequencies.

Measurements at the bridge

Nakamura also had troubles in the high frequency range: the resonances of his driver and detector seem to have influenced heavily the coupling in this frequency range. The graphs presented for a wide frequency band (up to 5 kHz) in Figure 2 are the velocity normalised by the fixed driving force measured at different point of the bridges of an upright piano assembled and tuned. This quantity corresponds to the admittance at the driving point. On the graphs (Figure 2), the resonances of the driver and detector are pointed out by a single arrow and a double arrow respectively. Above 1 kHz 1 , the mobility becomes suddenly much larger. Besides the level of resonances, this general mobility increase is, according to Nakamura, due to vibrations between ribs; in the high frequency, the ribs become the fixed edge and the inside board vibrates2 . However, Nakamura's (1980).

measurements need to be reconsidered in the high frequency range. Measurements done by Giordano [START_REF] Giordano | Mechanical impedance of a piano soundboard[END_REF]) (Figure 3) confirm this step-like falloff in the local impedance (or mobility increase) at high frequencies. Giordano notices that the step only occurs above approximately 2.5 kHz when the measurement is done at bridge.

It is interesting to notice that below the (first) impedance falloff (≈ 700 Hz), the average levels of the impedance measured at the bridge near a rib (1 to 2 • 10 3 kg s -1 ) and somewhere else on the soundboard between ribs (0.6 to 0.7 • 10 3 kg s -1 ) differ by a factor of 2 to 3, certainly due to the added stiffness by the bridge. The average low frequency impedance level measured at the bridge is comparable to Wogram's measurements.

Nightingale & Bosmans [START_REF] Nightingale | On the drive-point mobility of a periodic rib-stiffened plate[END_REF] studied the influence of the position of the driving point on the mobility of a periodic rib-stiffened isotropic plate. The space between the ribs was approximately 40 cm. The figure 4 points out that the real part of the mobility is a function of the distance to the nearest adjacent rib: the mobility decreases with the distance to an adjacent rib.

Besides, the mobility increases with frequency and tends to the When this ratio is less than ≈ 0.25 the influence of the ribs is large; the measured mobility is much less than that of the infinite plain plate.

These considerations explain why on the upright soundboard studied by Giordano, the impedance falloff between ribs appears at a much smaller frequency than when the impedance is measured on the treble bridge, close to a rib (≈ 700 Hz for the red curve and ≈ 2.5 kHz for the black curve, in figure 3). Moreover, in the light of the conclusions of Nigthingale et al., we can expect that the black and the red curves meet above 10 kHz, with a roughly constant impedance of 200 to 300 kg s -1 corresponding to the characteristic impedance of the infinite plain board for bending waves.

Conklin's measurements [START_REF] Conklin | Design and tone in the mechanoacoustic piano. Part 2. Piano structure[END_REF] are, to our opinion, the more accurate and reliable published measurements of a

Figure 5: Normalised mobility as a function of the normalised wave number in the guide (see text for normalising factors), after [START_REF] Nightingale | On the drive-point mobility of a periodic rib-stiffened plate[END_REF].

mechanical mobility at a piano bridge. Typical curves for the mobility normal to the soundboard are presented in figure 6. For the sake of comparison, we superpose two sets of measurements done for the same concert grand piano. The mobility when the strings and the plate have been removed appears in solid black line and the mobility at the same point when the instrument is fully assembled and tuned in dashed red line. When the metal frame and strings are added, the mobility curve is substantially altered. The frequencies of the first modes is increased while the peak values are about 15 dB less. This could mean that the modification of the structure has added damping. This effect can be considered as beneficial since it reduces fluctuations in mobility, as explained in previous section. Above 1 kHz the mobility is less modified. No measurements of the mobility in the direction normal to the soundboard have been published by Conklin above 3.2 kHz.

Conklin measured also the mobility at the bridge in the strings direction (see figure 7), with and without the frame and strings. Again two curves are superimposed in the figures: the mobility normal to the board (solid black line) and the "longitudinal mobility" (dashed blue line) measured at the same point. In the treble section (C 6 strings) and when the board is unloaded, the latter can be surprisingly larger than the former (by 10 to 20 dB) above ≈ 1 kHz (Fig. 7.a). The effect of the assembling on the longitudinal mobility is important: overall decrease of about 10-15 dB (the large longitudinal tension added by the strings to the bridge stiffens it and increases its longitudinal impedance).

Thus, for a piano in playing situation, the longitudinal mobility in treble region is comparable to the mobility in the direction normal to the board for frequencies between 2-3 kHz (Fig. 7.b).

It would be erroneous to ignore this mobility when dealing with the high-frequency tone of the piano sound. Askenfelt adds [START_REF] Askenfelt | Sound radiation and timbre[END_REF]): longitudinal string motion, which is known to influence the perception of bass notes, will thus be able to drive the soundboard rather efficiently in the high-frequency range.

SYNTHETIC DESCRIPTION

The purpose here is to give an expression of the piano soundboard mechanical mobility (in the direction normal to the soundboard) depending on a small number of parameters and valid up to several kHz.

Analytical expression: sum of the modal contributions

The driving point mobility Y A (at point (x A , y A )) of a weakly dissipative vibrating system can be expressed as the sum of the admittance of single-degree of freedom linear damped oscillators:

Y A (ω) = V A (ω) F A (ω) = iω +∞ ∑ ν=1 Φ 2 ν (x A , y A ) m ν (ω 2 ν + iη ν ω ν ω -ω 2 ) (1)
where m ν is the modal mass, η ν is the modal loss factor, ω ν the modal angular frequency and Φ ν the modal shape of the mode ν.

Skudrzyk mean-value theorem

The exact expression given above is useful to study the vibratory behaviour of a structure in the low-frequency domain where only a small number of parameters is sufficient to approximate the response of the structure (usually the sum is truncated at a pulsation between 3 and 10 times the pulsation of calculus). Higher in frequency, the detailed description becomes inapplicable since the number of needed parameters is too high. Instead, we present synthetic description of the mechanical mobility at bridge based on the Skudrzyk mean-value method [START_REF] Skudrzyk | The mean-value method of predicting the dynamic-response of complex vibrators[END_REF]). The method is quickly exposed here.

In the mid-and high-frequency domain, the frequency response of the structure tends to a smooth curve. The vibration can be described, ultimately, as a diffuse wavefield (see for example [START_REF] Skudrzyk | Vibrations of a system with a finite or an infinite number of resonances[END_REF] or [START_REF] Lesueur | Rayonnement acoustique des structures[END_REF]). Skudrzyk's idea, proposed in [START_REF] Skudrzyk | Vibrations of a system with a finite or an infinite number of resonances[END_REF][START_REF] Skudrzyk | Simple and complex vibratory systems[END_REF] and theorised in its final form in [START_REF] Skudrzyk | The mean-value method of predicting the dynamic-response of complex vibrators[END_REF] consists, in this spectral domain, in replacing the exact expression of the admittance (sum) (1) by an integral. By use of the residue theorem Skudrzyk calculates the integral and shows that the real part of the admittance in high frequency may be written as a function of the ratio of the modal density n and the mass of the structure M only, see (5). Because n and M are proportional to the surface S, the asymptotic value of the admittance depends neither on the excitation point nor on the surface: the structure can be considered as infinite in this frequency domain. This asymptotic value is naturally the characteristic admittance of the structure, noted Y C . By extrapolating towards the low frequencies, Skudrzyk's theory predicts the mean value and the envelope of the admittance:

G C = ℜ(Y C )
is the geometric mean of the values at resonances G res and antiresonances G ares . In summary, Skudrzyk's mean-value method predicts the envelope, the mean value and the asymptotic value of the driving point admittance of a weakly dissipative vibrating structure. Contrary to statistical methods (Statistical Energy Analysis (SEA) for example, see [START_REF] Lyon | Statistical Energy Analysis of Dynamical Systems : Theory and Applications[END_REF] e.g.), only valid in the high-frequency domain, this method gives indications on the mean behaviour of the structure from the first resonance up to the highest frequencies.

The principal results obtained by Skudrzyk are recalled here;

for the demonstrations the reader may refer to [START_REF] Skudrzyk | The mean-value method of predicting the dynamic-response of complex vibrators[END_REF].

The transformation of equation ( 1) into an integral is:

Y A (ω) → ω→+∞ Y C = +∞ 0 iωΦ 2 ν (x A , y A ) dω ν m ν ε ν (ω 2 ν + iη ν ω ν ω -ω 2 ) (2)
where

ε ν = dω ν dν = 2π ∆f ν = 2π n( f ν )
is the average modal spacing (written here for pulsations and corresponding to the inverse of the modal density n(ω ν )). The writing of the denominator of Y C can be simplified in the hypothesis of small damping.

For the oscillator ν, in the weakly dissipative case (η 2 ν 1), the damping term iη ν ω ν ω is negligible compared to ω 2 νω 2 for all ω except on the vicinity of the resonant frequency ω ν . The approximation introduced by [START_REF] Skudrzyk | Vibrations of a system with a finite or an infinite number of resonances[END_REF] or [START_REF] Cremer | Structure-Borne Sound[END_REF] consists in writing:

ω 2 ν + iη ν ω ν ω -ω 2 ≈ ω2 ν -ω 2 with ω2 ν = ω 2 ν (1 + iη ν )
(3) Given this, the equation ( 2) takes the form:

Y C = +∞ 0 iωΦ 2 ν (x A , y A ) dω ν m ν ε ν ( ω2 ν -ω 2 ) = G C + i B C ( 4
)
where

G C = ℜ(Y C ) and B C = ℑ(Y C ).
Finally, by use of the residue theorem, the real part of the driving point admittance is given by:

ℜ(Y A (ω)) → +∞ G C = π 2 ε ν M = n( f ) 4M (5)
In this frequency domain, the real part of the admittance depends only on the modal density and the mass of the structure. For a thin plate, the imaginary part B C vanishes at high frequency [START_REF] Skudrzyk | The mean-value method of predicting the dynamic-response of complex vibrators[END_REF]:

Y A (ω) → +∞ G C = 1 4h 2 3(1 -ν 2 xy ) E ρ (6)
written here in the isotropic case. G C is equivalent to the driving point admittance of the infinite plate [START_REF] Cremer | Structure-Borne Sound[END_REF]. It depends neither on the frequency, nor on the surface but only on the thickness h and on the elastic constants of plate: the Young's modulus E, the Poisson's ratio ν xy , and the density ρ.

Envelope

Skudrzyk gives an approached expression of the envelope of the resonances and antiresonances. Under the assumptions of well-separated peaks and equal modal masses (peaks of the impulses responses of equal amplitudes), a single-degree of freedom damped oscillator ν has an amplitude at resonance f ν of:

G res ≈ 1 ηω ν M = n( f ν ) 4M 2 π µ( f ν ) = G C β ( f ν ) (7) with β ( f ) = 2 π µ( f ) = 2 πn( f )η f
and where the indicator µ( f ) = n( f )η f is the modal overlap factor defined as the ratio between the half-power modal bandwidth and the average modal spacing. Generally, µ increases with frequency, and thus the amplitude of resonances decreases. In the theory of Skudrzyk, G C is the geometric mean value of the admittance (for all the frequencies) G C = (G res G ares ) 1/2 . This yields directly the amplitude of antiresonances:

G ares ≈ G C β ( f ν ) = n( f ν ) 4M π µ( f ν ) 2 (8)

Langley calculations

The equations above are valid for small modal overlaps. When the frequency increases, the contribution of the admittance of the neighbouring modes needs to be considered. In that purpose, [START_REF] Langley | Spatially averaged frequency-response envelopes for one-dimensional and 2-dimensional structural components[END_REF] modifies the calculus and evaluates analytically the envelope of the sum (1). He supposes that the resonances f ν are regularly spaced with an average modal spacing equal to the inverse modal density at the frequency of interest, that is for the resonance f p : f νf p = (νp)/n( f p ). Under this assumption, the envelope of resonances G res becomes:

G res ≈ G C µ( f p ) 2π ∞ ∑ j=1-p 1 j 2 + (µ( f p )/2) 2 (9)
It is possible to calculate this sum (by extending the lower limit on the summation to -∞):

G res ≈ G C coth π µ( f p ) 2 (10)
Similarly, supposing that the minima of admittance appear halfway between two successive resonances (that is at frequency

f = f ν + f ν+1 2 = f ν + 1 2 n( f ν )
), the envelope of antiresonances is given by:

G ares ≈ G C µ( f p ) 2π ∞ ∑ j=-∞ 1 ( j -1/2) 2 + (µ( f p )/2) 2 = G C tanh π µ( f p ) 2 (11)
For small modal overlaps, equations ( 10) and ( 11) established by Langley are equivalent to the ones given by Skudrzyk:

coth (π µ/2) ∼ 0 2/(π µ) = β ; tanh (π µ/2) ∼ 0 π µ/2 = β -1
For high frequencies, these two factors have the same limit (one) and the envelope tends to G c , which is consistent with the theory of Skudrzyk.

Irregular natural frequency spacing

Bidimensional structures, such as plates can present repeated resonances, degeneracy and thus irregular modal spacing. This severely degrades the accuracy of the admittance envelope given by ( 10) and ( 11). Langley introduces semi-empirical modifications in order to take into account these irregularities. The approach is based upon existing literature concerning statistical repartition of the resonances in room acoustics, [START_REF] Bolt | Note on normal frequency statistics for rectangular rooms[END_REF][START_REF] Bolt | Normal frequency spacing statistics[END_REF] or [START_REF] Sepmeyer | Computed frequency and angular distribution of normal modes of vibration in rectangular rooms[END_REF]. Under the assumption that the modal spacing conforms to the Poisson law, the amplitudes of resonant frequencies of a bi-dimensional rectangular structure are given by [START_REF] Langley | Spatially averaged frequency-response envelopes for one-dimensional and 2-dimensional structural components[END_REF]):

G res ≈ G C (1 + µ -1/2 2 ) coth (1 + µ -1/2 2 ) π µ 2 2 ( 12
)
where the modal overlap factor

µ 2 = [1 -(L 1 L 2 ) -1
] µ is modified to take into account the repeated frequencies. µ 2 depends on the natural numbers L 1 and L 2 related to the aspect ratio of the rectangular structure by L 2 /L 1 = L y /L x . The amplitude of antiresonances are:

G ares ≈ G C K tanh K π µ 2 2 (13)
where the factor K is given by the semi-empirical formula:

K =    1/2.3 si µ 2 < 1 1 + µ 2 - √ µ 2 2.3 + µ 2 -1 si µ 2 ≥ 1 (14)

APPLICATION ON AN UPRIGHT PIANO

The theory exposed in the previous section is now applied to the soundboard of an upright piano placed in a pseudo-anechoic room. The piano (see figure 8) is tuned normally but strings are muted by strips of foam inserted between the strings or by woven in two or three places. A particular attention is taken not to change the mechanical mobility at bridge. The modal behaviour of the soundboard is investigated by means of a recently published high-resolution modal analysis technique [START_REF] Ege | High-resolution modal analysis[END_REF]) avoiding the frequency-resolution limitations of the Fourier transform. The method of measurements, the signal processing treatments and the extraction of the modal density and mean loss factors (up to 2.5 kHz) are exposed in detail in our ICA companion-paper [START_REF] Ege | Vibrational and acoustical characteristics of the piano soundboard[END_REF] devoted to the vibration and some radiation characteristics of the piano soundboard. tance tends towards the theoretical asymptote, and the envelope given by the first calculus of Langley (10) coincides for the whole spectrum with the resonances and antiresonances of the syntesized admittance. The approximation of Skudrzyk is satisfactory only for frequencies less than ≈ 200 Hz, corresponding to a modal overlap smaller than 20% (average modal spacing more than five time the half-power modal bandwidth).

Secondly, we refined the model by considering now the structure as an isotropic rectangular plate, of constant thickness, of dimensions L x = 1.39 m, L y = 0.91 m and total mass M = 9 kg. Indeed, up to 1.1 kHz, our experimental and numerical investigations confirm previous results showing that the soundboard behaves like a homogeneous plate with isotropic properties and clamped boundary conditions (the mechanical characteristics of the homogeneous plate are given in [START_REF] Ege | Vibrational and acoustical characteristics of the piano soundboard[END_REF]).

For this calculus, the natural frequencies of the plate are calculated analytically (boundary conditions supposed to be simply supported). Thus, contrary to the previous case, the spectral repartition of the resonances is now irregular. The modal shapes are given by

Φ mn (x, y) = sin (k x m x) sin k y n y
with m and n natural numbers, and where the wave numbers in directions x and y are k x m = m π L x and k y m = n π L y . The modal masses are equal to M/4. We present on figure 10 the driving point admittance (equation ( 1)) synthesized for a point of the medium bridge: here in (4L x /5,5L y /6). The mean value Figure 10: Real part of the driving point admittance Y A ( f ) (-) synthesized for a point of the medium bridge of the piano soundboard studied and for simply supported boundary conditions. In order to take into account the irregularity of the modal spacing, the envelope of ℜ(Y A ) (-) is semi-empirically modified in (-• -) according to Langley,. At low frequencies, the mean value G C (-) deviates from the asymptote (--) due to the boundary conditions. of the mobility between 100 and 1000 Hz is approximately 1.3 • 10 -3 s kg -1 corresponding to an impedance of about 800 kg s -1 . This value is consistent with the measurements at the bridge published by [START_REF] Wogram | Acoustical Research on Pianos. Part I: Vibrational Characteristics of the Soundboard[END_REF] or [START_REF] Giordano | Mechanical impedance of a piano soundboard[END_REF]: these authors measured a mean impedance for typical upright piano of about 10 3 kg s -1 (see second section). Moreover, the fluctuations of the mobility for those frequencies are ±10-15 dB, which is also consistent with measurements published by [START_REF] Conklin | Design and tone in the mechanoacoustic piano. Part 2. Piano structure[END_REF] for example. Concerning the envelope, we observe that the first calculus by Langley underestimates the amplitudes of oscillations of the mobility. The semi-empirical modifications corrects partially the envelope that becomes satisfactory around 1 kHz.

INTER-RIB EFFECT -STRUCTURAL MODIFICA-TIONS

For frequency higher than 1100 Hz this simplest model is no more valid. Indeed, the half-wavelength at 1.1 kHz is equal to the average distance p between two consecutive ribs: ribs confine the wave propagation. The soundboard behaves as a set of waveguides in this spectral domain. This behaviour, already found by [START_REF] Berthaut | Piano soundboard: structural behavior, numerical and experimental study in the modal range[END_REF], is experimentally and numerically shown in our companion paper [START_REF] Ege | Vibrational and acoustical characteristics of the piano soundboard[END_REF]): for frequencies above 1.1 kHz, the modal density n( f ) measured on the soundboard falls significantly and the antinodes of vibration of the numerical modal shapes are localised between the ribs. A simple model of this bi-dimensional propagation media is developed from wich the modal density of the first transverse mode of the waveguide is derived. The latter takes the form (the complete calculus is presented in [START_REF] Ege | Vibrational and acoustical characteristics of the piano soundboard[END_REF]):

n(ω) = L y π √ 2Cω √ A 2 + 4Cω 2 -4B √ A 2 + 4Cω 2 -4B -A 1/2 (15)
where

A = D 2 + D 4 D 3 k 2 x m , B = D 1 D 3 k 4 x m ,C = ρ h D 3
and where the D i are the constants of rigidity of spruce, considered as an orthotropic material (of main axes x and y):

D 1 = E x h 3 /(12(1 -ν xy ν yx )), D 2 = ν yx E x h 3 /(6(1 -ν xy ν yx )), D 3 = E y h 3 /(12(1 -ν xy ν yx ))
and D 4 = G xy h 3 /3. E is the Young's modulus, ν is the Poisson's ratio, ρ the density, h the plate thickness and L y the length of the waveguide. The direction x is parallel to the grain of the spruce board and, thus, perpendicular to the ribs (see figure 8).

We extend now the approaches of Skudrzyk and Langley for this bi-dimensional media. The mean value of the driving point mobility in this spectral domain is given by the relation ( 5) where the modal density n and mass M are replaced now by the ones of the waveguide considered. Hence, for the soundboard studied, the mean value of the impedance would fall in theory from 800 kg s -1 before localisation of the waves to a value at 2500 Hz for example of 230 kg s -1 (about 3.5 times less). This value is calculated for the waveguide 2-3 (located between the second and third ribs) situated in the treble area of the instrument (upper left corner). This waveguide has a thickness h ≈ 8 mm, a length L y ≈ 55 cm, a width (inter-rib distance) p ≈ 12.8 cm and a modal density n ≈ 4 • 10 -3 modes Hz -1 at 2500 Hz.

The fall of impedance (rise of mobility) predicted by our synthetic description in the high-frequency range match previous observations by Wogram, Nakamura, Nightingale et al. or Giordano (see the bibliographical review given above). In particular, the value obtained theoretically for a typical waveguide is very close to the published measurements of Giordano for example (figure 3) who measured an impedance value of about 200 to 300 kg s -1 in the treble area of the instrument and between two consecutive ribs.

These results points out that the rise of mobility in this frequency range is directly linked to the inter-rib effect appearing when the half-wavelength becomes equal to the rib spacing. Thus the inter rib distance p appears as a fundamental parameter in the acoustic of the instrument. If this distance p is too large, the mobility at bridge will be too great in the treble range and the piano will exhibit less than normal durations of tones and a harsh sound. Conversely it p is too low (too much ribs on the board) the mobility level will be small, the duration longer than normal while, at the same time, the output will seem subnormal in the treble.

The previous paragraphs shows how the synthetic description can be used to predict the influence of a structural modification on the driving point mobility. Similarly, the modification of the thickness h of the waveguide but also of the material charac-teristics (Young's modulus E, density ρ) may be linked to the modal density and the mass of the propagation media and thus to the mean value of the driving point mobility, thanks to the synthetic description developed.

In order to go one step further in the analysis and envisage a possible improvement of the sound of the instrument, the construction of a soundboard on which the influence of these structural modifications may be directly measured is an absolutely necessity. We found only one experimental study, carried out by Conklin on a concert grand, where the influence of ribbing on the driving point mobility is investigated [START_REF] Conklin | Soundboard construction for stringed musical instruments[END_REF]. Conklin built a soundboard with 39 ribs (more than twice the usual number), reducing the spacing p to a value of ≈ 5 to 6 cm. With this value, the first cut-off frequency (when the wavelength λ = 2p) is raised at the highest frequency of Conklin's interest, that is in his study the fundamental of the highest string of the piano: C 8 ≈ 4186 Hz. The height of the ribs was the same as those of a normally-designed soundboard. Their width was changed to around 1.1 cm, approximately one half of the usual value, in order to keep almost the same stiffness and mass of the conventional board (the moment of inertia I rib of a rib that determines its stiffness is proportional to its width a but varies as the cube of its height b: I rib = ab 3 /12). In his own words, Conklin's new soundboard has improved uniformity of frequency response, improved and extended high frequency response, higher efficiency at higher frequencies, and improved tone quality. Nevertheless we believe that these conclusions need to be taken with precautions. No measure were published and the soundboard has not been commercialised. It presented surely some defects not reported by the author.

CONCLUSION

We have given an expression of the piano soundboard mechanical mobility (in the direction normal to the soundboard) depending on a small number of parameters and valid up to several kHz. This synthetic description is derived from Skudrzyk's and Langley's work: the mean value of the driving point mobility and its envelope are expressed with only the modal density n( f ), the mean loss factor η( f ) and the mass M of the structure. This theory is applied to an upright piano, from which the modal density and the modal loss factors were measured beforehand up to 2.5 kHz with a novel high-resolution modal analysis technique [START_REF] Ege | Vibrational and acoustical characteristics of the piano soundboard[END_REF]. The synthetised mechanical mobility at bridge matches experimental observations and could be used for numerical simulations for example. In particular it is shown that the evolution of the modal density with frequency is consistent with the rise of mobility (fall of impedance) in this frequency range and that both are due to the inter-rib effect appearing when the half-wavelength becomes equal to the rib spacing.

  Only four measurements of the admittance (or impedance) at the bridge of a piano soundboard have been published: Nakamura (1983) (Figure2) and[START_REF] Conklin | Design and tone in the mechanoacoustic piano. Part 2. Piano structure[END_REF] (Figure6) present the mobility at bridge whereas[START_REF] Wogram | Acoustical Research on Pianos. Part I: Vibrational Characteristics of the Soundboard[END_REF] (Figure1) and Giordano (1998) (Figure3) claim to have measured the impedance. All of these measurements are done in the direction normal to the soundboard and for upright pianos with muted strings. Conklin measured both the mobility normal to soundboard and the mobility in direction of strings of a concert grand.

Figure 1 :

 1 Figure 1: Magnitude of the impedance |Z| at bridge , and its phase ϕ measured at terminating point for strings of F 4 (key n • 46) of an upright piano in playing situation, after Wogram (1980).

( a )Figure 2 :

 a2 Figure 2: Mobility at bridge at different terminating point of an upright piano in playing situation, after Nakamura (1983).

Figure 3 :

 3 Figure 3: Comparison of the driving point impedance measured on an upright pianos in two different locations. -: at bridge (terminating point for strings of C 4 -key n • 40), after Giordano (1998). --: far away from the bridge, at a mid-point between two ribs.

Figure 4 :

 4 Figure 4: Real part of the driving point mobility measured in different points of a ribbed plate, as a function of the distance from a rib, after Nightingale and Bosmans (2006).

Figure 6 :

 6 Figure 6: Bridge mobility (direction normal to the soundboard) of a grand piano at the terminating point of the E 2 strings (key n • 20), after Conklin (1996). -: strings and plate removed. --: assembled and tuned.

  Figure 7: Comparison of the transverse and longitudinal mobilities between 0 and 3.2 kHz at the end point of the C 6 strings (key n • 64) of a grand piano, after Conklin (1996). -: mobility in the direction normal to the soundboard. --: mobility in the string direction.

Figure 8 :Figure 9 :

 89 Figure 8: Rear view of the upright piano studied.

Note that this value is the same in Wogram's measurements.

Nakamura adds in the same paper that he obtained Chladni patterns where vibrations between ribs are recognised, above 1.2 kHz. Unfortunately, these figures have not been published.

This approach avoids the detailed description of the soundboard, based on a very high number of parameters. Moreover the synthetic description can be used to predict the changes of the driving point mobility, and possibly of the sound radiation in the treble range, resulting from structural modifications (changes in material, geometry, average ribs spacing, etc.).


Arnaud Lagriffoul

Jean-Luc Boudenne

Rafik Absi

Jean-Jacques Ballet

Jean-Marc Berjeaud

Sylvie Chevalier

Edmond E Creppy

Eric Gilli

Bacterial-based additives for the production of artificial snow: What are the risks to human health?

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Tables


G F Thibaut

Agathe Keller

Overlooking mathematical justifications in the Sanskrit tradition: the nuanced case of

How did the narratives of the history of Indian mathematics explain the tradition of mathematical justifications that existed in medieval Sanskrit commentaries? When the German philologer G. F.Thibaut published a translation of a set of Vedic geometrical texts in 1874 and 1875, he established that India had known other mathematical activities than 'practical calculations'. Thibaut's philological work and historiographical values determined his approach to these texts and provided a bias for understanding the reasonings and efforts which establish the validity of algorithms in this set of texts.

Introduction

1 Until the 1990s, the historiography of Indian mathematics largely held that Indians did not use "proofs" 2 in their mathematical texts. Dhruv Raina has shown that this interpretation arose partly from the fact that during the second half of the nineteenth century, the French mathematicians who analyzed Indian astronomical and mathematical texts considered geometry to be the 1 I would like to thank K. Chemla and M. Ross for their close reading of this article.

They have considerably helped in improving it. 2 Srinivas 1990[START_REF] Hayashi | The Bakhshālī Manuscript. An ancient Indian mathematical treatise[END_REF].

1 measure of mathematical activity3 . The French mathematicians relied on the work of the English philologers of the previous generation, who considered the computational reasonings and algorithmic verifications merely 'practical' and devoid of the rigor and prestige of a real logical and geometrical demonstration. Against this historiographical backdrop, the German philologer Georg Friedrich Wilhem Thibaut (1848-1914) published the oldest known mathematical texts in Sanskrit, which are devoted only to geometry.

These texts, śulbasūtras4 (sometimes called the sulvasūtras) contain treatises by different authors (Baudhāyana, Āpastamba, Kātyāyana and Mānava) and consider the geometry of the Vedic altar. These texts were written in the style typical of aphoristic sūtras between 600 and 200 BCE. They were sometimes accompanied by later commentaries, the earliest of which may be assigned to roughly the thirteenth century. In order to understand the methods which he openly employed for this corpus of texts, Thibaut must be situated as a scholar. This analysis will focus on Thibaut's historiography of mathematics, especially on his perception of mathematical justifications.

1 Thibaut's intellectual background G. F. Thibaut's approach to the śulbasūtras combines what half a century before him had been two conflicting traditions. As described by D. Raina and F. Charette, Thibaut was equal parts acute philologer and scientist investigating the history of mathematics.

1.1 A philologer Commentaries can be hardly done without in the case of any Sanskrit astronomical work. . . However, Thibaut also remarks that because they were composed much later than the treatises, such commentaries should be taken with critical distance20 :

Trustworthy guides as they are in the greater number of cases, their tendency of sacrificing geometrical constructions to numerical calculation, their excessive fondness, as it might be styled, of doing sums renders them sometimes entirely misleading.

Indeed, Thibaut illustrated some of the commentaries' 'mis-readings' and devoted an entire paragraph of his 1875 article to this topic. Thibaut explained that he had focussed on commentaries to read the treatises but disregarded what was evidently their own input into the texts. Thibaut's method of openly discarding the specific mathematical contents of commentaries is crucial here. Indeed, according to the best evidence, the tradition of 'discussions on the validity of procedures'21 appear in only the medieval and modern commentaries. True, the commentaries described mathematics of a period different than the texts upon which they commented. However, Thibaut valued his own reconstructions of the śulbasūtras proofs more than the ones given by commentaries.

The quote given above shows how Thibaut implicitly values geometrical reasoning over arithmetical arguments, a fact to which we will return later.

It is also possible that the omission of mathematical justifications from the narrative of the history of mathematics in India concerns not only the conception of what counts as proof but also concerns the conception of what counts as a mathematical text. For Thibaut, the only real mathematical text was the treatise, and consequently commentaries were read for clarification but not considered for the mathematics they put forward.

In contradiction to what has been underlined here, the same 1875 article sometimes included commentator's procedures, precisely because the method they give is 'purely geometrical and perfectly satisfactory'22 . Thus there was a discrepancy in between Thibaut's statements concerning his methodology and his philological practice.

Thibaut's conception of the Sanskrit scholarly tradition and texts is also Through his belief in a contemptuous arrogance on the part of the writers, Thibaut implicitly denies the treatises any claim for reasonable mathematical justifications, as we will see later. Thibaut attributed part of the clumsiness which he criticized to their old age 26 :

23 See Dalmia 1996: 328 sqq. 24 Thibaut 1888: vii. I am setting aside here the fact that he argues in this introduction for a Greek origin of Indian astronomy.

25 [] indicate the author's addenda for the sake of clarity. 26 Thibaut 1875: 60.

Besides the quaint and clumsy terminology often employed for the expression of very simple operations (. . . ) is another proof for the high antiquity of these rules of the cord, and separates them by a wide gulf from the products of later Indian science with their abstract and refined terms.

After claiming that the treatises had a dogmatic nature, Thibaut extends this to the whole of "Hindu literature"27 :

The astronomical writers (. . . ) therein only exemplify a general mental tendency which displays itself in almost every department of Hindu Literature; but mere dogmatic assertion appears more than ordinarily misplaced in an exact science like astronomy. . . This "rougher and bolder treatment" is evidence, for instance, in his philologically accurate but somewhat clumsy translation of technical vocabulary. He thus translates dīrghacaturaśra (literally 'oblong quadrilateral')

variously; it is at some times a 'rectangular oblong', and at others an 'oblong' 29 . The expression 'rectangular oblong' is quite strange. Indeed, if the purpose is to underline the fact that it is elongated, then why repeat the idea? The first of Thibaut's translations seems to aim at expressing the fact that a dīrghacaturaśra has right-angles, but the idea of orthogonality is never explicit in the Sanskrit works used here, or even in later literature.

Thibaut's Historiography of Science

For Thibaut, 'true science' did not have a practical bent. In this sense, the science embodied in the śulbas, which he considered motivated by a practical religious purpose, is 'primitive' 30 :

The way in which the sūtrakāra-s [those who compose treatises]

29 See for instance, Thibaut 1875: 6. 30 Thibaut 1875: 17. . . . we must remember that they were interested in geometrical truths only as far as they were of practical use, and that they accordingly gave to them the most practical expression' Contemptuous as he may be of the state of Indian mathematics, Thibaut did not believe that the śulbasūtras were influenced by Greek geometry 37 .

For Thibaut, history of mathematics ought to reconstruct the entire deductive process from the origin of an idea to the way it was justified. Although later commentaries may include some useful information, they do not give us the key to understanding how these ideas were developed at the time when the treatises were composed. This lack of information provoked

Thibaut to complain about Indian astronomical and mathematical texts.

Thibaut clearly considered the texts to have been arranged haphazardly because the order of the rules do not obey generative logic. He thus defined his task 38 :

. . . I shall extract and fully explain the most important sūtra-s (. . . ) and so try to exhibit in some systematic order the knowledge embodied in these ancient sacrificial tracts

Here, Thibaut assumed that these works-not treatises but 'tracts' (presumably with derogatory connotations)-are not clear and systematic. Further, Thibaut felt the need to disentangle ('extract') the knowledge they contain.

In his view, this knowledge may be quite remarkable but it was ill presented. Thus commenting a couple years later on the Vedā ṅgajyotis . a, he remarked 39 :

37 Thibaut 1875: 4. This however will still be discussed as late as Staal 1999. 38 Thibaut 1875: 5. 39 Thibaut 1877:411, emphasis is mine.

The first obstacle in our way is of course the style of the treatise itself with its enigmatical shortness of expression, its strange archaic forms and its utter want of connection between the single verses.

He thus sometimes remarked where the rules should have been placed according to his logic. All the various texts of the śulbasūtras start by describing how to construct a square, particularly how to make a square from a rectangle.

However, Thibaut objected 40 :

their [the rules for making a square from a rectangle] right place is here, after the general propositions about the diagonal of squares and oblongs, upon which they are founded Consequently, Thibaut considered the śulbasūtras as a single general body of text and selected the scattered pieces of the process he hoped to reconstruct from among all the sūtras composed by various authors. At the same time, he distinguished the different authors of the śulbasūtras and repeatedly insisted that Āpastamba is more 'practical' than Baudhāyana, whom he preferred. For instance, an example of his method 41 :

Baudhāyana does not give the numbers expressing the length of the diagonals of his oblongs or the hypotenuses of the rectangular triangles, and I subjoin therefore some rules from Āpastamba, which supply this want, while they show at the same time the practical use, to which the knowledge embodied in Baudhāyana's sūtra could be turned

No mathematical justifications in the śulbasūtras

However, for Thibaut, Baudhyāna and probably other 'abstractly bent' treatise writers doubtlessly wanted to justify their procedures. More often than not, these authors did not disclose their modes of justification. Thus, when the authors are silent, Thibaut developed fictional historical procedures. For instance43 :

The authors of the sūtra -s do not give us any hint as to the way in which they found their proposition regarding the diagonal of a square [e.g. the Pythagorean proposition in a square] ;

but we may suppose that they, too, were observant of the fact that the square on the diagonal is divided by its own diagonals into four triangles, one of which is equal to half the first square.

Two different rules for a same result

Indeed, Thibaut underlined that several algorithms are occasionally given in order to obtain the same result. This redundancy puzzled him at times. For instance 46 , Thibaut examined the many various caturaśrakaran . a-methods to construct a square-given by different authors. Āpastamba, Baudhāyana and Kātyāyana each gave two methods to accomplish this task. I will not expose these methods here; they have been explained amply and clearly elsewhere 47 .

Thibaut also remarked that in some cases, Baudhāyana gives a rule and its reverse, although the reverse cannot be grounded in geometry. Such is the case with the procedure to turn a circle into a square 48 :

Considering this rule closer, we find that it is nothing but the reverse of the rule for turning a square into a circle. It is clear, however, that the steps taken according to this latter rule could not be traced back by means of a geometrical construction, for if we have a circle given to us, nothing indicates what part of the diameter is to be taken as the atiśayat . r . tīya (e.g. the segment of the diameter which is outside of the square) I am no specialist of śulba geometry and do not know if we should see the doubling of procedures and inverting of procedures as some sort of 'proofs', but at the very least they can be considered efforts to convince the reader that the procedures were correct. The necessity within the śulbasūtras to convinced and to verify has often been noted in the secondary literature, but has never fully or precisely studied 49 . Thibaut, although puzzled by the fact, never addressed this topic. Similarly, later historians of mathematics have noted that commentators of the śulbasūtras sought to verify the procedures while setting aside the idea of a regular demonstration in these texts. Thus Delire notes that Dvārakānātha used arithmetical computations as an easy method of verification (in this case of the Pythagorean Theorem) 50 . The use of two separate procedures to arrive at the same result, as argued in another article of this volume 51 , could have been a way of mathematically verifying the correctness of an algorithm-an interpretation that did not occur to Thibaut.

Conclusion

Thibaut, as we have thus seen, embodied contradictions. On the one hand, he swept aside the Sanskrit literary tradition and criticized its concise sūtras as obscure, dogmatic and following no logic whatsoever. On the other hand, as an acute philologer, he produced nuanced studies on the differences among the approaches of different authors. Through his naive assumption of a practical mind of the 'Hindu astronomers', his fruitless search for proper visual demonstrations in an algorithmic tradition, and a disregard of commentaries in favor of the treatises, Thibaut envisioned a tradition of mathematics in India blind to the logic that could have been used to justify the algorithms which he studied. Such arguments could have been perceived through the case of the 'doubled' procedures in the śulbasūtras, and maybe even through the arithmetical readings of these geometrical texts found in later commentaries. 50 Delire op. cit. : 129. 51 See Keller same volume.

  contradictory. He alternates between a vision of a homogenous and a-historicalIndian society and culture and the subtleties demanded by the philological study of Sanskrit texts.In 1884, as Principal of Benares Sanskrit College (a position to which he had been appointed in 1879), Thibaut entered a heated debate with Bapu Pramadadas Mitra, one of the Sanskrit tutors of the college, on the question of the methodology of scholarly Sanskrit pandits. Always respectful to the pandits who helped him in his work, Thibaut always mentioned their contributions in his publications. Nonetheless, Thibaut openly advocated a 'Europeanization' of Sanskrit Studies in Benares and sparked a controversy about the need for Pandits to learn English and history of linguistics and literature. Thibaut despaired of an absence of historical perspective in Pandits reasonings-an absence which led them often to be too reverent towards the past 23 . Indeed, he often criticized commentators for reading their own methods and practices into the text, regardless of the treatises' original intentions. His concern for history then ought to have led him lead to consider the different mathematical and astronomical texts as evidence of an evolution. However, although he was a promoter of history, this did not prevent him from making his own sweeping generalizations on all the texts of the Hindu tradition in astronomy and mathematics. He writes in the introduction of the Pañcasiddhānta 24 : (. . . ) these works [astronomical treatises by Brahmagupta and Bhāskarācarya] 25 claim for themselves direct or derived infallibility, propound their doctrines in a calmly dogmatic tone, and either pay no attention whatever to views diverging from their own or else refer to such only occasionally, and mostly in the tone of contemptuous depreciation.

  Thibaut does not seem to struggle with definitions of science, mathematics or astronomy, nor does he does discuss his competency as a philologer in undertaking such a study. In fact, Thibaut clearly states that subtle philology is not required for mathematical texts. He thus writes at the beginning of the Pañcasiddhānta 28 : . . . texts of purely mathematical or astronomical contents may, without great disadvantages, be submitted to a much rougher and bolder treatment than texts of other kinds. What interests us in these works, is almost exclusively their matter, not either their general style or the particular words employed, and the peculiar nature of the subject often enables us to restore with nearly absolute certainty the general meaning of passages the single words of which are past trustworthy emendation.

  Thibaut's translation, then, is not literal but colored by his own idea of what a dīrghacaturaśra is. Similarly, he calls the rules and verses of the treatises, the Sanskrit sūtras, 'proposition(s)', which gives a clue to what he expects of a scientific text, and thus also an inkling about what kind of scientific text he suspected spawned the sulbasūtras.

  enumerated above, must of course be imagined as a very primitive one. Nothing in the sūtra-s [the aphorisms with which treatises are composed] would justify the assumption that they were expert in long calculations.' However, he considered the knowledge worthwhile especially because it was geometrical 31 : It certainly is a matter of some interest to see the old ācārya-s [masters] attempting to solve this problem [squaring of the circle], which has since haunted so m[an]y unquiet minds. It is true the motives leading them to the investigation were vastly different from those of their followers in this arduous task. Theirs was not the disinterested love of research which distinguishes true science 32 , nor the inordinate craving of undisciplined minds for the solution of riddles which reason tells us cannot be solved; theirs was simply the earnest desire to render their sacrifice in all its particulars acceptable to the gods, and to deserve the boons which the gods confer in return upon the faithful and conscientious worshipper.' Or again 33 :

  practical aspect of these primitive mathematics explains why the methods they used were geometrical 34 : It is true that the exclusively practical purpose of the Śulvasūtra-s necessitated in some way the employment of practical, that means in this case, geometrical terms,. . . This geometrical basis distinguished the śulbasūtras from medieval or classical Indian mathematical treatises. Once again, Thibaut took this occasion to show how his preference for geometry over arithmetic 35 : Clumsy and ungainly as these old sūtra-s undoubtedly are, they have at least the advantage of dealing with geometrical operations in really geometrical terms, and are in this point superior to the treatment of geometrical questions which we find in the Līlāvatī and similar works.As made clear from the above quotation, Thibaut was a presentist historian of science who possessed a set of criteria which enabled him to judge the contents and the form of ancient texts. In another striking instance, Thibaut gives us a clue that Euclid is one of his references. Commenting on rules to make a new square of which the area is the sum or the difference of two known squares, Thibaut states in the middle of his own translation of Baudhāyana's śulbasūtras 36 : Concerning the methods, which the Śulvasūtras teach for caturasrasamāsa (sum of squares) and caturasranirhāra (subtraction of squares). I will only remark that they are perfectly legitimate; they are at the bottom the same which Euclid employs.

40

  Thibaut 1875: 28. 41 Thibaut 1875: 12. a long Sanskrit philological tradition. In other words, when a commentator extracts a new reading from one or several sūtras, he demonstrates the fruitfulness of the sūtras. The commentator does not aim to retrieve a univocal singular meaning but on the contrary underline the multiple readings the sūtra can generate. Additionally, as Thibaut rightly underlined, geometrical reasoning represented no special landmark of correctness in reasoning to medieval Indian authors. Because of these expectations and misunderstandings Thibaut was enable to find the mathematical justifications that maybe were in these texts. Let us thus look more closely at the type of reconstruction which Thibaut employed, particularly in the case of proofs 2 Practices and Readings in the History of Science It is telling that the word 'proof' is used more often by Thibaut in relation to philological reasonings than in relation to mathematics. Thus, as we have seen above, the word is used to indicate that the clumsiness of the vocabulary establishes the śulbasūtras's antiquity.

  This is at the same time an immediately convincing proof of the Pythagorean proposition as far as squares or equilateral rectangular triangles are concerned.' (. . . ) But how did the sūtrakāra-s [composers of treatises] satisfy themselves of the general truth of their second proposition regarding the diagonal of rectangular oblongs? Here there was no such simple diagram as that which demonstrates the truth of the proposition regarding the diagonal of the square, and other means of proof had to be devised.Thibaut thus implied that diagrams were used to 'show' the reasoning literally and thus 'prove' it. This method seems to hint that authors of the medieval period of Sanskrit mathematics could have had some sort of geometrical justification 44 . Concerning Āpastamba's methods of constructing fire altars, which was based on known Pythagorean triplets, Thibaut stated 45 :In this manner Āpastamba turns the Pythagorean triangles known to him to practical use (. . . ) but after all Baudhāyana's way of mentioning these triangles as proving his proposition about the diagonal of an oblong is more judicious. It was no practical want which could have given the impulse to such a research [on how to measure and construct the sides and diagonals of rectangles]-for right angles could be drawn as soon as one of the vijñeya [determined] oblongs (for instance that of 3, 4, 5) was known-but the want of some mathematical justifications which might establish a firm conviction of the truth of the proposition.So, in both cases, Thibaut represented the existence and knowledge of several Pythagorean triplets as the result of not having any mathematical justification for the Pythagorean Theorem. Thibaut proceeded to use this fact as a criterion by which to judge both Āpastamba's and Baudhāyana's use of Pythagorean triplets. Thibaut's search for an appropriate geometrical mathematical justification in the śulbasūtras may have made him overlook a striking phenomenon.

47

  Thibaut 1875: 28-30, Datta 1993: 55-62, Bag & Sen 1983 and finally Delire 2002:75 sqq. 48 Thibaut 1875: 35. 49 See for instance Datta 1993: 50-51.

  Thibaut trained according to the German model of a Sanskritist 5 . Born in 1848 in Heidelberg, he studied Indology in Germany. His European career culminated when he left for England in 1870 to work as an assistant for Max Müller's edition of the Vedas. In 1875, he became Sanskrit professor at Benares Sanskrit College. At this time, he produced his edition and studies of the śulbasūtras, the focus of the present article 6 . Afterwards, Thibaut This astronomical treatise written in the seventeenth century in Benares attempts to synthesize the re-workings of theoretical astronomy made by the astronomers under the patronage of Ulug Begh with the traditional Hindu siddhāntas 8 . Thibaut's next direct contribution to the history of mathematics and astronomy in India was a study on the medieval astronomical treatise, the Pañcasiddhānta of Varāhamihira. In 1888, he also edited and translated this treatise with S. Dvivedi and consequently entered in a heated debate with H. Sacred Books of the East, the series initiated by his teacher Max Müller 10 . Thibaut died in Berlin at the beginning of the

	Yajvan's commentary 13 on the Baudhāyana sulbasūtra and Rāma's (fl. 1447/1449)
	commentary on Kātyāyana's text. Thibaut also occasionally quotes Kapar-
	disvāmin's (fl. before 1250) commentary of Āpastamba 14 . Thibaut's intro-
	ductory study of these texts shows that he was familiar with the extant
	philological and historical literature on the subject of Indian mathematics
	and astronomy. However, Thibaut does not refer directly to any other schol-
	ars. The only work he acknowledges directly is A. C. Burnell's catalogue
	of manuscripts 15 . For instance, Thibaut quotes Colebrooke's translation of
	Līlāvatī 16 but does not refer to the work explicitly. Thibaut also reveals some
	general reading on the the history of mathematics. For example, he implic-
	itly refers to a large history of attempts to square the circle, but Thibaut's
	sources are unknown.
	His approach to the texts shows the importance he ascribed to acute philo-
	logical studies 17 . Thibaut often emphasizes how important commentaries are
	first world war, in October 1914. for reading the treatises 18 :
	Among the śulbasūtras, Thibaut focussed on Baudhāyana (ca. 600 BCE) 11 the sūtra-s themselves are of an enigmatical shortness (. . . ) but and Āpastamba's texts, occasionally examining Kātyāyana's śulbapariśis . t . a. the commentaries leave no doubt about the real meaning
	Thibaut noted the existence of the Mānavasulbasūtra but seems not to have
	had access to it 12 . For his discussion of the text, Thibaut used Dvārakānātha The importance of the commentary is also underlined in his introduction
	8 See Minkowski 2001 and CEES, Vol 2: 21. of the Pañcasiddhānta 19 :
	9 Thibaut 1899, Thibaut 1907. 13 Thibaut 1875: 3. 10 Thibaut 1904. 14 Thibaut 1877: 75. 11 Unless stated otherwise, all dates refer to the CESS. When no date is given, the CESS 15 Thibaut 1875: 3.
	likewise gives no date. 16 Thibaut 1875: 61.
	12 For general comments on these texts, see Bag & Sen 1983, CESS, Vol 1: 50; Vol 2: 17 See for instance Thibaut 1874: 75-76 and his long discussions on the translations of
	30; Vol 4: 252. For the portions of Dvārakānātha's and Venkateśvara's commentaries on vr . ddha.
	Baudhāyana's treatise, see Delire 2002 (in French).
	4

spent the following 20 years in India, teaching Sanskrit, publishing translations and editing numerous texts. With P. Griffith, he was responsible for the Benares Sanskrit Series, from 1880 onwards. As a specialist in the study of the ritualistic mimām . sa school of philosophy and Sanskrit scholarly grammar, Thibaut made regular incursions in the history of mathematics and astronomy.

Thibaut's interest in mathematics and astronomy in part derives from his interest in mimām . sa. The authors of this school commented upon the ancillary parts of the Vedas (vedā ṅga) devoted to ritual. The śulbasūtras can be found in this auxiliary literature on the Vedas. As a result of having studied these texts, between 1875 and 1878 7 , Thibaut published several articles on vedic mathematics and astronomy. These studies sparked his curiosity about the later traditions of astronomy and mathematics in the Indian subcontinent and the first volume of the Benares Sanskrit Series, of which Thibaut was the general scientific editor, was the Siddhāntatattvaviveka of Bhat . t . a Kamalākara.

Jacobi on the latter's attempt to date the Veda on the basis of descriptions of heavenly bodies in ancient texts. At the end of his life, Thibaut published several syntheses of ancient Indian mathematics and astronomy 9 . His main oeuvre, was not in the field of history of science but a three volume translation of one of the main mimām . sa texts: Śa ṅkarācārya's commentary on the Vedāntasūtras, published in the 18 op. cit. : 18.

19

Thibaut 1888: v.

See Raina 1999: chapter VI.

We will adopt the usual transliteration of Sanskrit words which will be marked in italics, except for the word Veda, which belongs also to English dictionaries.

The following paragraph rests mainly on Stachen-Rose 1990.

See Thibaut 1874[START_REF] Thibaut | On the Sulvasutras[END_REF], Thibaut 1877.

The last being a study of the jyotis . avedā ṅga, in Thibaut 1878.

Thibaut 1875: 61-62.

These are discussed, in a specific case, in the other article in this volume I have written, Keller same volume.

This concludes a description of how to transform a square into a rectangle as described by Dvāraka ṅtha in Thibaut 1875: 27-28.

Thibaut 1888: vii.

Thibaut 1888: v.

Thibaut 1875: 61.

Thibaut 1875: 60.

Thibaut 1877: 76.

Thibaut 1875: 11-12.

See Keller 2005. Bhāskara's commentary on the Āryabhat . īya was not published during Thibaut's lifetime, but I sometimes suspect that either he or a pandit with whom he worked had read it. The discussion on vis . amacaturaśra and samacaturaśra, in Thibaut 1875: 10, thus echoes Bhāskara I's discussion on verse 3 of chapter 2 of the Āryabhat . īya. Thibaut's conception of geometrical proof is similar to Bhāskara's as well.

Thibaut 1875: 17.

Thibaut 1875: 28-30.

When alternating among several authors was insufficient for his purposes, Thibaut supplied his own presuppositions. Indeed, Thibaut peppered his text with such reconstructions: 42 :

The authors of the sūtra -s do not give us any hint as to the way in which they found their proposition regarding the diagonal of a square; but we may suppose . . . (. . . ) The question arises : how did Baudhāyana or Āpastamba or whoever may have the merit of the first investigation, find this value? (. . . ) I suppose that they arrived at their result by the following method which accounts for the exact degree of accuracy they reached.' (. . . ) Baudhāyana does not state at the outset what the shape of his wheel will be, but from the result of his rules we may conclude his intention. etc.

Because he had an acute idea of what was logically necessary, Thibaut thus had a clear idea of what was sufficient and insufficient for reconstructing the processes. As a result, Thibaut did not deem the arithmetical reasoning of Dvārakānātha adequate evidence of mathematical reasoning.

The misunderstandings on which Thibaut's judgments rest are evident.

For him, astronomical and mathematical texts should be constructed logically and clearly, with all propositions regularly demonstrated. This presumption compelled him to overlook what he surely must have known from his familiarity with Sanskrit scholarly texts: the elaborate character of a sūtra-marked by the diverse readings that one can extract from it-enjoyed 42 Thibaut 1875: 11, 18, 49.


Agathe Keller

Dispelling Mathematical Doubts. Assessing Mathematical Correctness of Algorithms in Bhāskara's Commentary on the Mathematical Chapter of the Āryabhat . īya

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction 1

Contrary to the perception prevalent at the beginning of the twentieth century, a concern for the mathematical correctness of algorithms existed 1 I would like to thank Karine Chemla and Micah Rosss for their attentive and helpful scrutinizing of this article.

1 in the mathematical tradition in Sanskrit. Reflections on the systematic upapattis of Kr . s . n . a's (fl. ca. 1600Kr . s . n . a's (fl. ca. -1625) ) commentary on the Bījagan . ita, the explorations of the Mādhava school (fourteenth-sixteenth century) or other traditions of mathematical validity have already been published. 2 Still, the variations among this tradition of justification and explanation need to be studied.

In the following sections, the Āryabhat . īyabhās . ya of Bhāskara (BAB) is analyzed with regard to its reasoning and vocabulary. The second chapter of Āryabhat . īya (Ab)-an astronomical siddhānta composed in verse at the end of the fifth century-treats mathematics (gan . ita). Respecting the requirements of the genre, these aphoristic āryas usually provide the gist of a procedure, such as an essential relationship or the main steps of an algorithm. The BAB is not only the earliest known commentary on this treatise but also the oldest known text of mathematics in Sanskrit prose that has been handed down to us. The BAB thus gives us a glimpse into the reasonings used in the scholarly mathematical tradition in Sanskrit at the beginning of the seventh century 3 . Very little is known about who practiced scholarly mathematics in classical India, and why scholarly texts were elaborated. The BAB provides information on the intellectual context in which both the Ab and the BAB were composed. First, the commentator's defense of Āryabhat . a's treatise (and the commentator's own interpretations of the verses) will provide a backdrop for reflections on the mathematical correctness of procedures. Next, the arguments behind the algorithms of mathematical justification will be 2 Ikeyama 2003, Jain 2001, Patte 2004, Srinivas 1990. Some of Kr . s . n . a's demonstrations are noted, among others, in the footnotes of Colebrooke 1817.

3 Keller 2006, Introduction. clarified. Afterwards, Bhāskara's vocabulary including explanations, proofs and verifications will be more precisely characterized.

1 Defending the Treatise Bhāskara's commentary, a prolix prose text, gives us a glimpse into the intellectual world of scholarly astronomers and mathematicians. The commentary records their intellectual debates. For the opening verse in which the author of the treatise mentions his name, Bhāskara's commentary explains:

. . . as a heroic man on battle fields, whose arms have been copiously lacerated by the strength of vile swords, having entered publicly a battle with enemies, who proclaims the following, as he kills: 'This Yajñadatta here ascended, a descendant of the Aditis, having undaunted courage in battle fields, now strikes.

If someone has power, let him strike back!' In the same way, this master also, who has reached the other side of the ocean of excessive knowledge about Mathematics, Time-reckoning and the Sphere, having entered an assembly of wise men, has declared: ' Āryabhat . a tells three: Mathematics, Time-reckoning, the Sphere'. Within this hostile atmosphere, Bhāskara's commentary attempts to convince the reader of the coherence and validity of Āryabhat . a's treatise. To this end, the commentary dispels 'doubts' (sandeha) that arise in the explanations of Āryabhat . a's verses. Thus, the analysis provides refutations (parihāra) to objections and establishes (sādhya, siddha) Bhāskara's readings of Āryabhat . a's verse. This commentary presents mainly syntactical and grammatical discussions which debate the interpretation of a given word in the treatise. More often than not, the discussion of the meaning and use of a word defines and characterizes the mathematical objects in question. (Are squares all equal sided quadrilaterals? Do all triangles have equally halving heights? etc.). 5 Bhāskara's commentary adopts technical words, and the specialized readings of the verses show that the Ab cannot be understood in a straightforward way. The verses need interpretation and the interpretation should be the correct one.

The search for the proper interpretation thus defines the commentator's task. The importance of interpretation becomes especially clear when Bhāskara criticizes Prabhākara's exegesis of the Ab. 6 For instance, in his comment on the rule for the computation of sines, Bhāskara explains that the expression samavr . tta refers to a circle, not a disk as Prabhakara understood it. 7 More crucially, through his understanding of the word agra trīn . i gadati gan . itam kālakriyām golam' iti uktavān /. question nonetheless is raised: Bhāskara argues that the procedures of the Āryabhat . īya are correct, but how does Bhāskara "establish" a rule? Moreover, what does Bhāskara consider a "proof" ? The answer to these questions presents difficulties. Indeed, the rational behind the fragmentary arguments that BAB sets forth is at times hard to grasp. The point of this aritcle is to show that two specific commentarial techniques, the 're-interpretation' of procedures and establishing an alternative independent procedure, were used to ground the Ab's rules. To establish this point, a characterization of these commentarial techniques will be necessary. This characterization will be followed by a description of the different ways Bhāskara explicitly tries to establish the mathematical validity of Ab's rules.

'Re-interpretating' of Procedures

Rule of Three

The Rule of Three (trairāśika 12 ) appears in verse Ab.2.26. Now, when one has multiplied that fruit quantity of the Rule of Three by the desire quantity|

The quotient of that divided by the measure should be this fruit of desire 13

In other words, if M (the measure) produces a fruit F M , and D is a desire for which the fruit, F D , is sought, the verse may be expressed in modern algebraic notation as:

F D = F M × D M (1)
12 For a general overview on the Rule of Three in India see Sarma 2002. 13 trairāśikaphalarāśim . tam athecchārāśinā hatam . kr . tvā| labdham . pramān . abhajitam . tasmād icchāphalam idam . syāt (Shukla 1976: 112-223).

7

Obviously, this expression can also be understood as a statement that the ratios are equal:

F D D = F M M (2)
The procedure given in the verse provides an order for the different operations to be carried out. First, the desire is multiplied by the fruit. Next, the result is divided by the measure. This order of operations causes the procedure to appear as an arbitrary set of operations14 . Bhāskara provides a standard expression to define the kind of problem which the Rule of Three solves. When the commentator thinks that a situation involves proportional quantities and thus the Rule of Three is (or can be) applied, he brings this fact to light by using a verbal formulation (vāco yukti ) of the Rule of Three.

This verbal formulation is a syntactically rigid question which reads as follows:

If the measure produces the fruit, then with the desire what is produced? The fruit of desire is produced.

This question, when it appears, shows that Bhāskara thinks that the Rule of Three can be applied. I believe that for Bhāskara the Rule of Three invokes proportionality. concerned by his reasoning. Two examples of Bhāskara's 're-interpretation' will demonstrate how he employed this theorem.

The Pythagorean Theorem

'Re-interpretation' with gnomons

The section devoted to gnomons (śa ṅku) contains two illuminating cases.

A Gnomon and a Source of Light

The standard situation is as follows: A gnomon (śa ṅku, DE) casts a shadow (EC), produced by a source of light (A), as illustrated in Figure 2. This procedure involves a multiplication and a division. In modern algebraic notation:

EC = BE × DE AF
The procedure given in the verse appears to be an arbitrary set of operations. Bhāskara begins with a general gloss. Then, as in all his verse commentaries, Bhāskara's commentary provides a list of solved examples.

These examples have a standard structure: first comes a versified problem, then a 'setting down' (nyāsah . ) section, and finally a resolution (karan . a).

Thus, in his 're-interpretation' of the above procedure after a solved example , Bhāskara writes:

This computation is the Rule of Three. How? If from the top of the base which is greater than the gnomon [AF ], the size of the space between the gnomon and the base, which is a shadow, The segment SuS'u illustrates the Rsine of altitude. Bhāskara I notes that the triangle SuS'uO is similar to GOC. Therefore the segment S'uO (that is, the Rsine of the zenithal distance) is proportional to the shadow of the gnomon at noon and the Rsine of the altitude is proportional to the length of the gnomon. This propotionality is further illustrated in Figure 4. Indeed, as illustrated in Figure 5, on the equinoxes the sun is on the celestial equator. At noon, the sun occupies the intersection of the celestial Shukla 1976:90. 21 This is an approximate value. For more on this value, see Keller 2006, BAB.2.14. 22 This value is also an approximation.

[F D = BE] is obtained, then,
as the base of a right triangle (bhujā) and he identifies the radius of the celestial sphere as the hypotenuse. Thus, the Rsine is identified with the up-right side of a right-triangle. This identification implicitly explains how the computation is carried out. However, Bhāskara immediately adds 23 :

With the Rule of Three also 13,12,3438; what has been obtained

is the Rsine of the colatitude, 3174 24 .

In this way, Bhāskara again computes OS u by using the similarity of OSuSu and OGC. Bhāskara thus computes the same value twice, using two different methods. The most likely explanation is that he verifies the results obtained with one algorithm by using another independent process.

The mathematical key to both these computations is the prior relation- Shukla 1976, 90. 24 This value is an approximation again.

unexplained steps of the procedure are given a mathematical grounding that may serve as a justification of the algorithm itself.

This analysis thus brings to light two kinds of reasoning: the confirmation of a result by using two independent procedures and the mathematical grounding of a set of operations via their 're-interpretation' according to the Rule of Three and/or the 'Pythagoras Theorem'. These kinds of mathematical reasoning are also found in the parts of BAB which explicitly have a persuasive aim, attempting to convince the reader that the algorithms of the Ab are correct.

Explanations, Verifications and Proofs

Bhāskara uses specific names when referring to a number of arguements:

'explanations', 'proofs' (upapatti) and 'verifications' (pratyāyakaran . a). These arguments do not appear systematically in each verse commentary and -as will be seen below -are always fragmentary. The following description of explanations, proofs and verifications will attempt to highlight how they are structured and the different interpretations they can be subject to.

Explanations

Bhāskara's commentary on verse 8 of the mathematical chapter of the Ab presents an example of explanation. Verse 8 describes two computations concerning a trapezoid. (See Figure 3.1.) The first calculation evaluates the length of two segments (svapātalekha, EF and FG) of the height of a trapezoid. In this case, the height is bisected at the point of intersection In other words, with the labels used in Figure 3.1, we have:

EF = AB × EG AB + CD ; F G = CD × EG AB + CD .
Likewise, the area A is:

A = EG × (AB + CD) 2 .

25

āyāmagun . e pārśve tadyogahr . te svapātalekhe te| vistarayogārdhagun . e jñeyam . ks . etraphalam āyāme (Shukla 1976: 63).

On the first part of the verse, Bhāskara comments 26 :

The size of the 'lines on their own fallings' should be explained This explanation consists of 're-interpreting' the procedure-which is a multiplication followed by a divison-according to the Rule of Three. The explanation contains two steps. The first step considers the proportionality in a diagram, then 're-interprets' the set of operations of the algorithm as the application of the Rule of Three. As previously, the seemingly arbitrary set of operations is endowed with a mathematical meaning.

The second computation in verse 8 determines the area of the trapezoid.

As shown in Figure 3.1, the area of the trapezoid can be broken into the summation of the areas of several triangles. Alternately, the trapezoid can be decomposed into a rectangle and two triangles.

Here, with a previous rule [Ab.2.6.ab] the area of isosceles and uneven trilaterals should be shown/explained (darśayitavya). Or, with a rule which will be stated [Ab.2.9.] the computation of the area of the inner rectangular field should be performed ;

Even though it has not survived, such a figure shows how areas can be added to give the area of the trapezoid. This time, a collection of already known procedures, those computing the area of triangles and rectangles, is mobilised. We do not know if they are used to 're-interpret' the procedure or to establish an alternative independant procedure. The procedures of Ab.2.9 will be analyzed below.

Both of Bhāskara's explanations in BAB.2.8 consist of 1. an explanation of a diagram, and 2. either a 're-interpretation' of the procedure or exposing an indepent alternative procedure This 're-interpration' either confirms or verifies the reasoning by looking at a diagram.

Three words refer to an explanation: vyākhyāna, pradarśan . a, and pratipādita.

The word vyākhyāna indicates that the commentary gives an explanation, but it is also used for an argument connected with a diagram 28 :

Or else, all the procedures used in the production of chords with an explanation (vyakhyāna). Therefore it has not 29 been put forth (pratipādita) by Āryabhat . a in the Āryabhat . īya .

Note that this passage emphasizes that explanations belong to the genre of commentary and, at least according to Bhāskara, should not be exposed in a treatise.

The word pradarśan . a is derived from the verbal root dr . s . -, 'to see'. It has a similar range of meaning as the English verb 'to show'. It is often hard to distinguish if the word refers to the visual part of an explanation or to the entirety of the explanation. For instance, in BAB.2.11, Bhāskara uses a diagram and writes 30 :

In the field drawn in this way all is to be shown/explained (pradarśayitavya).

Finally, the word pratipādita is more technical and straightfoward. It commonly appears in lists of solved examples found in most of the commented verses in the mathematical part of BAB.

In the illustrations of explanations presented above, the commentator 're-interprets' geometrical procedures according to the Rule of Three or the 'Pythagorean Theorem'. Only geometrical procedures receive such arguments. Each time, the commentary omits a diagram to which the text seemingly refers. Among the geometrical processes, explanations are 'seen', as

29 na has been added by the editor, K. S. Shukla and is not found in the manuscripts.

Another possible interpretation of the sentence reads: 'Therefore it has been put forth by Bhāskara in his commentary 30 evam ālikhite ks . etre sarvam . pradarśayitavyam (Shukla 1976: 79).

22

will be seen in the only example from the BAB in which the word 'proof' occurs.

3.2

The Only Two Occurrences of the Word "Proof "

The Sanskrit word upapatti refers directly to a logical argument. This word is used twice in Bhāskara's commentary as noted by Takao Hayashi 31 . The gender of this word is feminine and it is derived from the verbal root upa-PAD-, meaning 'to reach'. Thus, an upapatti, is literally 'what is reached' and has consequently been translated as 'proof'. In both instances, some ambiguity surrounds this word, and the meaning of the word is not certain.

One occurrence has been quoted above, wherein proofs (upapatti) are described as opposed to tradition. The other instance refers to the reasoning whereby the height of a regular tetrahedron is determined from its sides. In this case, Bhāskara understands Āryabhat . a's rule in the second half of verse 6 of the mathematical chapter as the computation of the volume of a regular tetrahedron. Such a situation is described in Figure 7.

Given a regular tetrahedron ABCD, AH is the line through A perpendicular to the plane defined by the triangle BDC. AH is called the 'upward-side' (ūrdhvabhujā). AC is called karn . a (lit. ear) because it is the hypotenuse of A verification resembles an explanation in that a verification 're-interprets' a given procedure according to another rule and establishes a mathematical grounding. The arguments that the commentator labels "verifications" sometimes present difficulties, and currently our understanding of them is not at all certain. Below are set out several hypotheses about how these verifications can be understood.

Verification of an Arithmetical Computation

Bhāskara states a verification by the Rule of Five for the rule given in Ab.2.25.

Āryabhat . a states the rule in Ab.2.25 as follows 34 :

The interest on the capital, together with the interest on the interest , with the time and capital for multiplier, increased by the square of half the capital|

The square root of that, decreased by half the capital and divided by the time, is the interest on one's own capital This passage can be formalized as follows: Let m (mūla) be capital; let p 1 (phala) be the interest on m during a unit of time, k 1 = 1 (kāla), usually 33 Hayashi 1995:73-74.

34 mūlaphalam . saphalam . kālamūlagun . am ardhamūlakr . tiyuktam| tanmūlam . mūlārdhonam . kālahr . tam . svamūlaphalam (Shukla 1976: 114).

a month. Let p 2 be the interest on p 1 at the same rate for a period of time k 2 . If p 1 + p 2 , m, and k 2 are known, the rule can be expressed in a modern mathematical notation as:

p 1 = mk 2 (p 1 + p 2 ) + ( m 2 ) 2 -m 2 k 2 .
This rule is derived from a constant ratio:

m p 1 = p 1 p 2 k 2 .
The Rule of Five, described in BAB.2.26-27.ab, rests on the same ratio as the rule given in Ab.2.25. In the former instance though, k 1 may be a number other than 1:

m p 1 k 1 = p 1 p 2 k 2 .
The Rule of Five indicates an expression equal in value to p 2 :

p 2 = p 2 1 k 2 mk 1
The Rule of Five may therefore be used in the opposite direction to find a value for p 1 .

In The Rule of Five, as seen above, returns the value of p 2 . This procedure does not deliver the same result but gives a method of inverting the procedure to check independently that the result makes sense. In this case, an independent procedure is established. The use of the Rule of Five, which Bhāskara describes as a combination of two Rules of Three, also imbues the computation with a mathematical basis in proportionality.

Verification of the Area of Plane Figures

Bhāskara interprets the first half of Ab.2.9 as a way to verify procedures for areas given by Āryabhat . a in the previous verses.

For all fields, when one has acquired the two sides, the area Hayashi has interpreted this argument in such a manner. 39 The reasoning would produce a rectangle and verify that its area is equal to the area of the figure.

A third approach relies on the 'setting down' parts which contain diagrams. Such a verification consists of constructing a rectangle with the same area from a given figure. For instance, in the second step of the verification of the area of a triangle, Bhāskara specifies that when the parts of the area of such a triangle are rearranged (vyasta), they produce the rectangle which is drawn. The construction of a rectangle from the original figure is not described in Bhāskara's commentary. However, such constructions could have been known, as shown by the methods exposed in BAB.2.13. Further- 39 Hayashi 1995: 73. more, this process recalls the algorithms from the sulbasūtras, the earliest known texts of Sanskrit geometry. These algorithms produce a construction which, although not described in the text, corresponds with the discussion contained in the text. With just such a diagram, the argument in the text would arithmetically verify that the construction is correct.

These three interpretations can be combined if a verification is allowed to be simultaneously geometrical and arithmetical. Bhāskara relies on a geometrical strategy to produce a rectangle with the appropriate area, showing that he knows how to construct the corresponding rectangle from the initial figure. Because the construction is obvious, it would not be detailed, and only the lengths of the rectangle would be given. Regardless of which interpretation is accepted, the verification either 'reinterprets' a first algorithm (BAB.2.9) and produces a new understanding of the procedure, or it produces a new procedure that gives the same result (BAB.2.25). In either case, the so-called 'verification' confirms the numerical results and places the procedure in a secure mathematical context. Thus, after verification, the calculations do not appear to be a set of arbitrary steps.

Conclusion

This survey of the BAB has brought to light two kinds of reasonings checking the Ab rules and seeking to convince readers of their validity. One argument exhibits an independent alternative procedure. In one case the procedure exhibited arrives at the same result as the opposite direction procedure. The second reasoning, we have called it 're-interpretation', uses the Rule of Three and the 'Pythagoras Theorem' to provide a new outlook onto the arbitrary steps of the procedure. How should the Rule of Three and the so-called 'Pythagoras Theorem' be described in this context? They are mathematical tools which enable astronomical situations or specific problems to be 're-interpreted' as abstract and general cases, involving right triangles and proportionalities. The arbitrary steps of the procedure are thus given a mathematical explanation.

Nonetheless, the methods of reasoning are hard to understand and pin down. This difficulty may arise from their oral nature, of which Bhāskara's written text preserves only a portion. For instance, the function of diagrams in these reasonings still remains mysterious. Further detailed explorations of how Sanskrit texts explain, prove and verify mathematical algorithms will advance understanding about how the mathematical corectness of algorithms was conceptualized by mathematicians in the Indian subcontinent.

Bibliography

Colebrooke, H. T. (1817), Algebra, with Arithmetic and mensuration. London : J. Murray.

  Figure 2.2. In the first half of Ab.2.17, Bhāskara states the Pythagorean

Figure 2 :

 2 Figure 2: A Schematized Gnomon and Light

  2.3.2 A Gnomon inRelation to the Celestial Sphere In the previous commented verse (BAB.2.14), Bhāskara sets out two procedures. Both rest on the proportionality of the right triangle formed by the gnomon and its midday shadow with the right triangle composed by the Rsine of the altitude and the zenithal distance. In the present example, one procedure uses only the Rule of Three, while the other uses the Rule of Three with the Pythagoras Theorem. Both procedures compute the same results. Consider Figure 3. Here, GO represents a gnomon and OC indicates its midday shadow. The circle of radius OSu (Su symbolising the sun) represents the celestial meridian. The radius OSu is thus equal to the radius of the celestial sphere. S'u designates the projection of the Sun onto the horizon.

Figure

  Figure 3: Proportional Astronomical Triangles

Figure 4 :

 4 Figure 4: Altitude and Zenith

  equator and the celestial meridian. At that moment, the zenithal distance z equals the latitude of the gnomon (φ) and the altitude (a) becomes the co-latitude (90 -φ). Once again, the similarity of SuS uO and OGC is underlined by a certain number of puns. Here, the Rsine of latitude (SuSu ) is called 'perpendicular' (avalambaka). Now, Bhāskara considers an example for an equinox in which OG = 13, OC = 5, and the radius of the celestial sphere (SuO) is the customary 1348. Bhāskara writes 20 : When computing the Rsine of latitude (aks . ajyā) the Rule of Three is set down: 13, 5, 3438. What is obtained is the Rsine of latitude, 1322 21 . That is the base (bhujā) the half-diameter is the hypotenuse (karn . a); the root of the difference of the squares of the base and the hypotenuse is the Rsine of co-latitude (avalambaka), 3174 22In this case, Bhāskara uses the fact that the triangles are both right and similar. Bhāskara then uses this similarity to compute SuS u. Bhāskara employs the 'Pythagorean Theorem' to compute OS u. In order to identify the right triangle, Bhāskara renames the Rsine of latitude (aksajyā, SuS u) 20 aks . ajyā "nayane trairāśikasthāpanā-13/ 5/ 3438/ labdham aks . ajyā 1322/ es . ā bhujā, vyāsārdha karn . ah . , bhujākarn . avargaviśes . amūlam avalambakah . 3174.

  ship between the gnomon and the celestial sphere. A syntactical connection establishes the relationship between these two spaces. The invocation of the Rule of Three begins with a standard question. The naming of two of its segments identifies a right triangle. This identification indicates not only one of the mathematical properties underpinning the procedure but also maps the specific astronomical problem onto a more general and abstract mathematical situation. (That is, Rsines of altitudes and zenithal distances become the legs of a simple right-triangle.) Since this mathematical interpretation is linked to a set of operations (first multiplication and division, then squaring the lengths with subsequent additions or subtractions of the results), the 23 trairāśikenāpi 13/ 12/ 3438/ labdham avalambakah . 3174/

Figure 6 :

 6 Figure 6: Inner segments and fields in a trapezoid

(

  figure drawn by a person properly instructed. Then, by means of just the Rule of Three with both sides, a computation of the lines whose top is the intersection of the diagonals and a perpendicular is performed .

AHC.

  Bhāskara explains how to compute the upright side by using the Pythagorean Theorem and the Rule of Three. The determination of CH, from which the upright side AH may be computed, rests upon the proportional properties of similar triangles, illustrated in Figure8. The triangles31 Hayashi 1985: 75-76.

Figure 7 :

 7 Figure 7: An Equilateral Pyramid with a Triangular Base

is their product | 38

 38 Bhāskara endows the verse with the goal of 'verification'-a goal nowhere explicitly appearing in the verse itself. Two steps can be distinguished in the verifications of this verse commentary, each corresponding to a diagram.The first step constructs a diagram of the figure for which an area is verified.The length and width of a rectangle with the same area as the figure are identified. This 'length' and 'width' are usually values from Āryabhat . a's procedure for which verification is sought. For instance, to verify the area of a triangle, the length of the corresponding rectangle is identified as the height of the triangle, while the width of the rectangle is half the base of the triangle. Precisely, the area of a triangle is given elsewhere by Ab-in the first half of verse 6-as the product of half the base by the height of a triangle. 38 sarves . ām . ks . etrān . ām . prasādhya pārśve phalam . tadabhyāsah . | Shukla 1976: 66. 29 The second step of the argument presents a diagram of the rectangle and computes the multiplication. How should this argument be understood? According to one means of understanding, this argument is a formal interpretation. The reasoning would consist of considering the rule one seeks to verify as the multiplication of two quantities. Each quantity is then interpreted geometrically as either the length or width of a rectangle with the same area as the initial figure. In this way, Bhāskara calculates the length and height of the rectangle, as required by verse 9. Another way of understanding the argument begins with the fact that the verification for a given figure produces a rectangle of the same area as the given figure. The fact that all figures have a rectangle with the same area would then become an implicit assumption of Sanskrit plane geometry. T.

  From an arithmetic perspective, this 're-interpretation' provides a new understanding of the rule given by Āryabhat . a. Through his arithmetical 'verification,' Bhāskara explains the geometrical verification. Bhāskara explains the link between the sides of the initial figure and the lengths and widths of the rectangle with the same area as the initial figure.

  The standard formulation of the Rule of Three, applied to the similar triangles AFD and DEC, can be recognized here. The standard expression of the Rule of Three provides the proportional elements on which the computation is based. Here the rule indicates that the ratio of AF to FD is equal to the ratio of DE to EC. The 're-interpretation' of the rule thus gives the arbitrary set of operations a mathematical significance. Rather than just a list of operations, the rule in Ab.2.15 becomes a Rule of Three.

	17
	etatkarma trairāśikam/ katham ?	sa ṅkuto 'dhikāyā uparibhujāyā yadi
	śa ṅkubhujāntarālapramān . am . chāyā labhyate tadā śa ṅkunā keti chāyā labhy-
	ate

what is obtained with the gnomon [DE]? The shadow [EC] is obtained 17 . 16 śa ṅkugun . am . śa ṅkubhujāvivaram . śa ṅkubhujayor viśes . ahr . tam| yal labdam . sā chāyā jñeyā śa ṅkoh . svamūlāt hi Shukla 1976:90.

  are in the realm of a diagram, and a diagram is intelligible only

	pūrvasūtren . ātra	dvisamavis . amatryaśraks . etraphalam .	darśayitavyam/
	vaks . yān . asūtren . āntarāyatacaturaśraks . etraphalānayanam (. . . ) vā/
	Shukla 1976: 63.
	28 athavā jyotpattau yatkaran . am .	tatsarvam .	chedyakavis . ayam . , chedyakam .	ca

vyākhyānagamyamiti na pratipāditam/

(Shukla 1976: 79)

.

  BAB.2.25 Bhāskara gives an example 35 : By the procedure given in Ab.2.25, the value of p 1 is 5.Simply stated, the verification consists of knowing m, p 1 and k 2 , finding p 2 and confirming that its value increased by p 1 will give the same value for p 1 + p 2 as stated in the problem.

		Obtained in four months is six. State the interest of a hundred
			produced within a month
	This example states a case in which:
				m =	100
				k 2 =	4
				p 1 + p 2 = 6
		Bhāskara then adds 36 :
		Verification (pratyayakaran . a) with the Rule of Five :"If the monthly
		interest (vr . ddhi 37 ) on a hundred is five, then what is the interest of
		the interest [of value (dhana)-five] on a hundred, in four months?"
				1 4
		Setting down:	100 5	The result is one. This increased by the
				5 0
		monthly interest on the capital is six rūpas, 6.
		1. I do not know the monthly-interest on a hundred. However,
	36
			the monthly-interest on a hundred increased by the interest
		on the interest | atyayakaran . am pañcarāśikena yadi śatasya māsikī vr . ddhih . pañca tadā
		caturbhih . māsaih . śatavr . ddheh . [pañcadhanasya] kā vr . ddhih . iti/ nyāsah . -
	35	1	4
		jānāmi śatasya phalam na ca kintu śatasya yatphalam saphalam | māsaih . 100 5 labdham 1 / etatsahitā śatavr . ddhih . s . ad . rūpān . i 6/
		caturbhih . āptam s . ad . vada vr . ddhim śatasya māsotthām 5 0

If the division was made first (resulting in the "fruit" of one measure) and then the multiplication, the computation would have had a step-by-step meaning, but this is not the order adopted by Ab.

From now on, unless otherwise stated this is the word translated as 'interest'.

BB : CB = CB : CH.

From this relationship it is known that:

Bhāskara expressed this relationship as the Rule of Three. The text does not give a precise argument, but it alludes to the properties as being clear from a diagram. It is in this context that the word upapatti appears 32 :

In order to show the proof (upapatti) of that Rule of Three, a field is set-down.

The argument implied by this word depends on the diagram. As in the case of the explanations, the proof must have been presented orally. This situation differs from the acts of 're-interpretation' seen above. In the present case, an argument is created, and there is no pre-existing algorithm to 'reinterpret'. However, the foundations of this new arguement are set out in a diagram. Furthermore, the procedure used is the Rule of Three, as in the 'explanations' seen above. Another type of argument concerns the correctness of algorithms: verification.

Verification

Verifications are distinguished from explanations and proofs by their name, pratyayakaran . a. Indeed, pratyaya has an etymological root in a verb meaning 'to come back', which has connotations of conviction. Pratyayakaran . a 32 trairāśikopapttipradarśanārtham . ks . etranyāsah . - (Shukla 1976: 59).