text:
Translation:Theoria combinationis observationum erroribus minimis obnoxiae

1.
No matter how careful one is with observations concerning the measurement of physical quantities, they are inevitably subject to errors of varying degrees. These errors, in most cases, are not simple but arise from several distinct sources that it is best to distinguish into two classes.

Some causes of errors depend, for each observation, on variable circumstances independent of the result obtained: the errors arising from these are called "irregular" or "random," and like the circumstances that produce them, their value is not amenable to calculation. Such are the errors that arise from the imperfection of our senses and all those due to irregular external causes, e.g. vibrations of the air that blur our vision. Some of the errors due to the inevitable imperfection of even the best instruments, e.g. the roughness of the inner part of a level, its lack of absolute rigidity, etc., belong to this same category.

On the other hand, there are other causes that produce an identical error in all observations of the same kind, or one whose magnitude depends only on circumstances that can be viewed as essentially connected to the observation. We will call errors of this category "constant" or "regular" errors.

Moreover, one can see that this distinction is to a certain extent relative, and has a broader or narrower sense depending on the meaning one attaches to the idea of observations of the same nature. E.g. if one indefinitely repeats the measurement of the same angle, the errors arising from imperfect division of the instrument belongs to the class of constant errors. If, on the other hand, one successively measures several different angles, the errors due to imperfect division will be considered random until a table of errors relative to each division has been formed.

2.
We exclude the consideration of regular errors from our discussion. It is up to the observer to carefully investigate the causes that can produce a constant error, to eliminate them if possible, or at least assess their effect in order to correct it for each observation, which will then give the same result as if the constant cause had not existed. It is quite different for irregular errors: by their nature, they resist any calculation, and they must be tolerated in observations. However, by skillfully combining results, their influence can be minimized as much as possible. The following investigation is devoted to this most important topic.

3.
Errors arising from a simple and determinate cause in observations of the same kind are confined within certain limits that could undoubtedly be assigned if the nature of this cause were perfectly known. In most cases, all errors between these extreme limits must be considered possible. A thorough knowledge of each cause would reveal whether all these errors have equal or unequal likelihood, and in the latter case, what the relative probability of each of them is. The same remark applies to the total error resulting from the combination of several simple errors. This error will also be confined between two limits, one being the sum of the upper limits, the other the sum of the lower limits corresponding to the simple errors. All errors between these limits will be possible, and each can result, in an infinite number of ways, from suitable values attributed to the partial errors. Nevertheless, it is possible to assign a larger or smaller likelihood for each result, from which the law of relative probability can be derived, provided that the laws of each of the simple errors are assumed to be known, and ignoring the analytical difficulties involved in collecting all of the combinations.

Of course, certain sources of error produce errors that cannot vary according to a continuous law, but are instead capable of a finite number of values, such as errors arising from the imperfect division of instruments (if indeed one wants to classify them among random errors), because the number of divisions in a given instrument is essentially finite. Nevertheless, if it is assumed that not all sources of error are of this type, then it is clear that the complex of all possible total errors will form a series subject to the law of continuity, or, at least, several distinct series, if it so happens that, upon arranging all possible values of the discontinuous errors in order of magnitude, the difference between a pair of consecutive terms is greater than the difference between the extreme limits of the errors subject to the law of continuity. In practice, such a case will almost never occur, unless the the instrument is subject to gross defects.

4.
Let $$\varphi(x)$$ denote the relative likelihood of an error $$x{:}$$ this means, due to the continuity of the errors, that $$\varphi(x)\,{d}x$$ is the probability that the error lies between the limits $$x$$ and $$x + {d}x.$$ In practice it is hardly possible, or perhaps impossible, to assign a form to the function $$\varphi$$ a priori. Nevertheless, several general characteristics that it must necessarily present can be established: $$\varphi(x)$$ is obviously a discontinuous function; it vanishes for all values of $$x$$ not between the extreme errors. For any value between these limits, the function is positive (excluding the case indicated at the end of the previous article); in most cases, errors of opposite signs will be equally possible, and thus we will have: $$\varphi(x) = \varphi(-x)$$ Finally, since small errors are more easily made than large ones, $$\varphi(x)$$ will generally have a maximum when $$x = 0$$ and will continually decrease as $$x$$ increases.

In general, the integral $$\int_a^b \varphi(x) \, {d}x$$ expresses the probability that the unknown error falls between the limits $$a$$ and $$b.$$ It follows that the value of this integral taken between the extreme limits of the possible errors will always be $$=1.$$ And since $$\varphi(x)$$ is zero for values not between these limits, it is clear that in all cases

5.
Let us consider the integral $$\int_{-\infty}^{\infty} x\, \varphi(x) \, {d}x$$ and denote its value by $$k.$$ If the sources of error are such that there is no reason for two equal errors of opposite signs to have unequal likelihood, we will have $$\varphi(x) = \varphi(-x),$$ and consequently, $$k = 0.$$ We conclude that if $$k$$ does not vanish and has e.g. a positive value, then there necessarily exists an error source that produces only positive errors or, at least, produces them more easily than negative errors. This quantity $$k,$$ which is the average of all possible errors, or the average value of $$x,$$ can conveniently be referred to as the "constant part of the error". Moreover, it is easily proven that the constant part of the total error is the sum of the constant parts of the simple errors of which it is composed.

If the quantity $$k$$ is assumed to be known and subtracted from the result of each observation, then, denoting the error of the corrected observation by $$x'$$ and the corresponding probability by $$\varphi'(x'),$$ we will have $$x' = x - k,$$ $$ \varphi'(x') = \varphi(x),$$ and consequently, $$\textstyle \int x'\varphi'(x')\,{d}x' = \int x\,\varphi(x)\,{d}x - k\int \varphi(x)\,{d}x = k - k = 0,$$ i.e. the errors of the corrected observations will have no constant part, which is clear in and of itself.

6.
The value of the integral $$\int_{-\infty}^{\infty} x \varphi(x) \, {d}x,$$ which is the average value of $$x,$$ reveals the presence or absence of a constant error, as well as the value of this error. Similarly, the integral $$\int_{-\infty}^{\infty} x^2 \varphi(x) \, {d}x,$$ which is the average value of $$x^2,$$ seems very suitable for defining and measuring, in a general manner, the uncertainty of a system of observations. Therefore, between two systems of observations of unequal precision, the one giving a smaller value to the integral $$\int_{-\infty}^{\infty} x^2 \varphi(x) \, {d}x$$ should be considered preferable. If it is argued that this convention is arbitrary and seemingly unnecessary, then we readily agree. The question at hand is inherently vague and can only be delimited by a somewhat arbitrary principle. Determining a quantity through observation can be likened, somewhat accurately, to a game in which there is a loss to be feared and no gain to be expected; each error being likened to a loss incurred, the relative apprehension about such a game should be expressed by the probable loss, i.e., by the sum of the products of the various possible losses by their respective probabilities. But what loss should be likened to a specific error? This is not clear in itself; its determination depends partly on our whim. It is evident, first of all, that the loss should not be regarded as proportional to the error committed; for, in this hypothesis, a positive error representing a loss, the negative error should be regarded as a gain: on the contrary, the magnitude of the loss should be evaluated by a function of the error whose value is always positive. Among the infinite number of functions that fulfill this condition, it seems natural to choose the simplest one, which is undoubtedly the square of the error, and thus we are led to the principle proposed above.

considered the question in a similar manner, but adopted as a measure of loss the error itself, taken positively. This assumption, if we do not deceive ourselves, is no less arbitrary than ours: should we, indeed, consider a double error as more or less regrettable than a simple error repeated twice, and should we, consequently, assign it a double or more than double importance? This is a question that is not clear, and on which mathematical arguments have no bearing; each must resolve it according to their preference. Nevertheless, it cannot be denied that Laplace's assumption deviates from the law of continuity and is therefore less suitable for analytical study; ours, on the other hand, is recommended by the generality and simplicity of its consequences.

7.
Let us define $$\int_{-\infty}^{\infty} \varphi(x) \, x^2 \, {d}x = m^2{:}$$ we will call $$m$$ the "mean error to be feared" or simply the "mean error" of the observation whose indefinite errors $$x$$ have a relative probability of $$\phi x.$$ We do not limit this designation to the immediate result of the observations, but rather extend it to any quantity that can be derived from them in any way. It is important not to confuse this mean error with the arithmetic mean of the errors, which is discussed in art. 5.

When comparing several systems of observations or several quantities resulting from observations that are not given the same precision, we will consider their relative "weight" to be inversely proportional to $$m^2,$$ and their "precision" to be inversely proportional to $$m.$$ In order to represent the weights by numbers, we should take, as the unit, the weight of a certain arbitrarily chosen system of observations.

8.
If the errors of the observations have a constant part, subtracting it from each obtained result reduces the mean error, increases the weight and precision. Retaining the notation of art. 5, and letting $$m'$$ denote the mean error of the corrected observations, we have

If, instead of the constant part $$k,$$ another number $$l$$ were subtracted from each observation, the square of the mean error would become

9.
Let $$\lambda$$ be a determined coefficient and let $$\mu$$ the value of the integral

Then $$\mu$$ will be the probability that the error of a certain observation is less than $$\lambda m$$ in absolute value, and $$1 - \mu$$ will be the probability that this error exceeds $$\lambda m.$$ If, for $$\mu = \tfrac{1}{2},$$ $$\lambda m$$ has the value $$\rho,$$ it will be equally likely for the error to be smaller or larger than $$\rho{:}$$ thus, $$\rho$$ can be called the probable error. The relationship between $$\lambda$$ and $$\mu$$ depends on the nature of the function $$\varphi,$$ which is unknown in most cases. However,i t is interesting to study this relationship in some particular cases.

I. If the extreme limits of the possible errors are $$+a$$ and $$-a,$$ and if, between these limits, all errors are equally probable, the function $$\varphi(x)$$ will be constant between these same limits, and, consequently, equal to $$\tfrac{1}{2a}.$$ Hence, we have $$m = a\,\sqrt{\tfrac{1}{3}},$$ and $$\mu = \lambda \,\sqrt{\tfrac{1}{3}},$$ so long as $$\lambda$$ is less than or equal to $$\sqrt{3};$$ finally $$\rho = m \sqrt{\tfrac{3}{4}} = 0{,}8660254 m,$$ and the probability that the error does not exceed the mean error is $$\sqrt{\tfrac{1}{3}} = 0{,}5773503.$$

II. If as before $$-a$$ and $$+a$$ are the limits of possible errors, and if we assume that the probability of these errors decreases from the error $$0$$ onwards like the terms of an arithmetic progression, then we will have

From this, we deduce that $$m = a\,\sqrt{\tfrac{1}{6}}$$ and $$\mu = \lambda\,\sqrt{\tfrac{2}{3}} - \tfrac{1}{6}\lambda^2,$$ as long as $$\lambda$$ is between 0 and $$\sqrt{6};$$ $$\lambda = \sqrt{6} - \sqrt{6 - 6\mu}$$ as long as $$\mu$$ is between 0 and 1; and finally,

In this case, the probability that the error remains below the mean error is

III. If we assume the function $$\varphi(x)$$ to be proportional to $$e^{-\frac{x^2}{h^2}},$$ then it must be equal to

where $$\pi$$ denotes the semiperimeter of a circle of radius $$1,$$ from which we deduce

(see Disquisitiones generales circa seriem infinitam, art. 28). If we let $$\Theta z$$ denote the value of the integral

then we have

The following table gives some values of this quantity:

10.
Although the relationship between $$\lambda$$ and $$\mu$$ depends on the nature of the function $$\varphi,$$ some general results can be established that apply to all cases where this function does not increase with the absolute value of the variable $$x;$$ then we have the following theorems:
 * $$\lambda$$ will not exceed $$\mu\sqrt{3}$$ whenever $$\mu$$ is less than $$\tfrac{2}{3};$$
 * $$\lambda$$ will not exceed $$\frac{2}{3\sqrt{1-\mu}}$$ whenever $$\mu$$ exceeds $$\tfrac{2}{3};$$

When $$\mu = \tfrac{2}{3},$$ the two limits coincide and $$\lambda$$ cannot exceed $$\sqrt{\tfrac{4}{3}}.$$

To prove this remarkable theorem, let $$y$$ be the value of the integral $$\int_{-x}^{+x} \varphi(z) \, {d}z.$$ Then $$y$$ will be the probability that an error is between $$-x$$ and $$+x.$$ Let us set

then we have $$\psi(0) = 0,$$ and

and by hypothesis $$\psi'(y)$$ is always increasing between $$y=0$$ and $$y=1,$$ or at least is not decreasing, or equivalently $$\psi''(y)$$ is always positive, or at least not negative. Now we have

thus,

Therefore, $$y\,\psi'(y)- \psi(y)$$ always has positive value, or at least this expression is never negative, and therefore

will always be positive and less than unity. Let $$f$$ be the value of this difference for $$y=\mu;$$ since $$\psi(\mu) = \lambda\,m,$$ we have

This being prepared, let's consider the function

which we set $$= {F}(y),$$ and also $${d}{.}{F}(y) = {F}'(y) \, {d}y.$$ Then it is clear that

Since $$\psi'(y)$$ is continually increasing with $$y$$ (or at least does not decrease, which should always be understood), and at the same time $${F}'(y)$$ is constant, the difference

will be positive for all values of $$y$$ greater than $$\mu,$$ and negative for all values of $$y$$ smaller than $$\mu.$$ It follows that the difference $$\psi(y) - {F}(y)$$ is always positive, and consequently, $$\psi(y)$$ will certainly be greater than $${F}(y)$$ in absolute value, as long as the function $${F}(y)$$ is positive, i.e. between $$y = \mu f$$ and $$y=1.$$ The value of the integral

will therefore be less than that of the integral

and a fortiori less than

i.e., less than $$m^2.$$ Now the value of the first of these integrals is found to be

and therefore $$\lambda^2 $$ is less than $$ \frac{3 \mu^2 (1-f)^2}{(1- \mu f)^3},$$ with $$f$$ being a number between $$0$$ and $$1.$$ If we consider $$f$$ as a variable, then this fraction, whose differential is

will be continually decreasing as $$f$$ increases from $$0$$ to $$1$$ so long as $$\mu$$ is less than $$ \tfrac{2}{3},$$ and therefore its maximum value will be found when $$f=0$$ and will be $$=3\mu^2,$$ so that in this case, the coefficient $$\lambda$$ will certainly be less, or at least not greater than $$\mu\sqrt{3}.$$ Q.E.P. On the other hand, when $$\mu$$ is greater than $$\frac{2}{3},$$ the maximum value of the function will be found when $$2 - 3\mu + \mu f = 0,$$ i.e. for $$f = 3 - \tfrac{2}{\mu},$$ and this maximum value will be $$=\frac{4}{9(1-\mu)},$$ so in this case, the coefficient $$\lambda$$ will not be greater than $$\frac{2}{3\sqrt{1-\mu}}.$$ Q.E.S.

Thus e.g. for $$\mu = \tfrac{1}{2}$$ it is certain that $$\lambda$$ will not exceed $$\sqrt{\tfrac{3}{4}},$$ which means that the probable error cannot exceed $$0{,}8660254 m,$$ to which it was found to be equal in the first example in art. 9. Furthermore, it is easily concluded from our theorem that $$\mu$$ is not less than $$\lambda\sqrt{\tfrac{1}{3}}$$ when $$\lambda$$ is less than $$\sqrt{\tfrac{4}{3}},$$ and on the other hand, it is not less than $$1 - \tfrac{4}{9\lambda^2}$$ when $$\lambda$$ is greater than $$\sqrt{\tfrac{4}{3}}.$$

11.
Since several of the problems discussed below involve the integral $$\textstyle \int x^4 \varphi(x) \, {d}x,$$ it will be worthwhile for us to evaluate it in some special cases. Let us denote the value of the integral

by $$n^4.$$ I. When $$\varphi(x) = \tfrac{1}{2a}$$ for values of $$x$$ between $$-a$$ and $$+a,$$ we have $$n^4 = \tfrac{a^4}{5} = \tfrac{9}{5}\,m^4.$$

II. In the second case of art. 9, with $$x$$ still between $$-a$$ and $$+a,$$ we have $$n^4 = \tfrac{1}{15}\,a^4 = \tfrac{12}{5}\,m^4.$$

III. In the third case, where

we find, as explained in the commentary cited above, that $$n^4 = \tfrac{3}{4}\,h^4 = 3\,m^4.$$

It can also be demonstrated, with only the assumptions of the previous article, that the ratio $$\tfrac{n^4}{m^4}$$ is never less than $$\tfrac{9}{5}.$$

12.
Let $$x,$$ $$x',$$ $$x'',$$ etc. denote the errors made in observations of the same kind, and suppose that these errors are independent of each other. Let $$\varphi(x)$$ be the relative probability of error $$x,$$ and let $$y$$ be a rational function of variables $$x,$$ $$x',$$ $x'',$ etc. Then the multiple integral

extended to all values of the variables $$x,$$ $$x',$$ $x,$ etc. for which the value of $$y$$ falls between the given limits $$0$$ and $$\eta,$$ represents the probability that the value of $$y$$ is between $$0$$ and $$\eta.$$ This integral is evidently a function of $$\eta,$$ whose differential we set $$= \psi(\eta)\,{d}\eta,$$ so that the integral in question is equal to $$\int_0^\eta \psi(\eta) \, {d}\eta,$$ and therefore, $$\psi(\eta)$$ represents the relative probability of an arbitrary value of $$y.$$ Since $$x$$ can be regarded as a function of the variables $$y,$$ $$x',$$ $x,$ etc., which we set the integral (I) will be

where $$y$$ takes values between $$y=0$$ and $$y=\eta,$$ and the other variables take all values for which $$f(y, x', x'', \ldots)$$ is real. Hence we have

the integration, where $$y$$ is to be regarded as a constant, being extended to all values of the variables $$x',$$ $x,$ etc. for which $$f(y, x', x, \ldots)$$ takes a real value.

13.
The previous integration would require knowledge of the function $$\varphi,$$ which is unknown in most cases. Even if this function were known, the calculation would often exceed the capabilities of analysis. Therefore, it will be impossible to obtain the probability of each value of $$y;$$ but it is different if one asks only for the average value of $$y,$$ which will be given by the integral $$\textstyle \int y \, \psi(y) \, {d}y,$$ extended to all possible values of $$y.$$ And since it is evident that for all values which $$y$$ cannot attain, either due to the nature of the function (e.g. for negative values, if $$ y = xx + x'x'+xx+$$ etc.), or because of the limits imposed on $$x,$$ $$x',$$ $x'',$ etc., one can assume that $$\psi(y)=0,$$ it is clear that the integration can be extended to all real values of $$y$$ from $$-\infty$$ to $$+\infty.$$

But the integral $$\int y \, \psi(y) \, {d}y,$$ taken between determinate limits $$\eta$$ and $$\eta',$$ is equal to the integral , taken from $$y = \eta$$ to $$y = \eta'$$ and extended to all values of the variables $$x',$$ $x'',$ etc. for which $$f$$ is real. This integral is therefore equal to the integral

in which $$y$$ is expressed as a function of $$x,$$ $$x',$$ $x'',$ etc., and the integration is extended to all values of the variables that leave $$y$$ between $$\eta$$ and $$\eta'.$$ Thus, the integral}}

can be obtained from the integral

where the integration is extended to all real values of $$x,$$ $$x',$$ $$x'',$$ that is, from $$x = -\infty$$ to $$x = +\infty,$$ $$x' = -\infty$$ to $x' = +\infty,$ etc.

If the function $$y$$ reduces to a sum of terms of the form

then the value of the integral

extended to all values of $$y,$$ or equivalently the average value of $$y,$$ will be equal to a sum of terms of the form

that is, the average value of $$y$$ is equal to a sum of terms derived from those that make up $$y,$$ by replacing $$x^\alpha,$$ $$x'^\beta,$$ $x''^\gamma,$ etc. with their average values. The proof of this important theorem could easily be derived from other considerations.

15.
Let us apply the theorem of the previous article to the case where

and $$\sigma$$ denotes the number of terms in the numerator.

We immediately find that the average value of $$y$$ is equal to $$m^2,$$ the letter $$m$$ having the same meaning as above. The true value of $$y$$ may be lower or higher than its average, just as the true value of $$x^2$$ may, in each case, be lower or higher than $$m^2;$$ but the probability that by chance, the value of $$y$$ differs by a small amount from $$m^2$$ will approach certainty as $$\sigma$$ becomes larger. In order to clarify this, since it is not possible to determine this probability exactly, let us investigate the mean error to be feared when $$y = m^2.$$ It is clear from the principles of art. 6 that this error will be the square root of the average value of the function

To find it, it suffices to observe that the average value of a term such as $$\frac{x^4}{\sigma^2}$$ is equal to $$\frac{n^4}{\sigma^2}$$ ($$n$$ having the same meaning as in art. 11), and that the average value of a term such as $$\frac{2{x^2}{x'}^2}{\sigma^2}$$ is equal to $$\frac{2m^4}{\sigma^2};$$ therefore, the average value of this function will be

Since this last formula contains the quantity $$n,$$ if we only want to get an idea of the precision of this determination, it will suffice to adopt a certain hypothesis about the function $$\varphi.$$ E.g. if we take the third assumption of arts. 9 and 11, this error will be equal to $$m^2 \sqrt\tfrac{2}{\sigma}.$$ Alternatively, we can obtain an approximate value of $$n^4$$ by means of the errors themselves, using the formula

In general, it can be stated that a precision twice as great in this determination will require a quadruple number of errors, meaning that the weight of the determination is proportional to the number $$\sigma.$$

Similarly, if the errors of the observations contain a constant part, we will deduce from their arithmetic mean a value of the constant part, and this value will be approached as the number of errors increases. In this determination, the mean error to be feared will be represented by $$\sqrt\frac{m^2 - k^2}{\sigma},$$ where $$k$$ denotes the constant part, and $$m$$ denotes the mean error of the observations uncorrected for their constant error. It will be simply represented by $$\frac{m}{\sqrt\sigma}$$ if $$m$$ represents the mean error of the observations corrected for the constant part (see art 8).

16.
In the arts. 12-15, we assumed that the errors $$x,$$ $$x',$$ $x'',$ etc. belonged to the same type of observation, so that the probability of each of these errors was represented by the same function. However, it is clear that the general principles outlined in arts. 12-14 can be applied with equal ease in the more general case where the probabilities of the errors $$x,$$ $$x',$$ $x,$ etc., are represented by different functions $$\varphi(x),$$ $$\varphi'(x'),$$ $$\varphi(x'')$$ etc., i.e. when these errors belong to observations of varying precision or uncertainty. Let $$x$$ denote the error of an observation with a mean error to be feared of $$m,$$ and let $$x',$$ $x,$ etc. denote the errors of other observations with mean errors to be feared of $$m',$$ $m,$ etc. Then the average value of the sum $$x^2 + {x'}^2 + {x}^2 +$$ etc. will be $$m^2 + {m'}^2 + {m}^2 + $$ etc. Now, if it is also known that the quantities $$m,$$ $$m',$$ $m,$ etc. are respectively proportional to the numbers $$1,$$ $$\mu',$$ $\mu,$ etc., then the average value of the expression

will be $$=m^2.$$ However, if we adopt for $$m^2$$ the value that this expression will take, by substituting the errors $$x,$$ $$x',$$ $x'',$ etc., as chance offers them, then the mean error affecting this determination will become, just as in the preceding article,

where $$n',$$ $n,$ etc., have the same meaning with respect to the second and third observation, as $$n$$ does with respect to the first; and if we can assume the numbers $$n,$$ $$n',$$ $n,$ etc., proportional to $$m,$$ $$m',$$ $m'',$ etc., this mean error to be feared will be equal to

But this method of determining an approximate value for $$m$$ is not the most advantageous. Consider the more general expression

whose average value will also be $$m^2,$$ regardless of the coefficients $$\alpha',$$ $\alpha,$ etc. The mean error to be feared when substituting the value $$m^2$$ for a value of $$y,$$ as determined by the likelihoods of $$x,$$ $$x',$$ $x,$ etc., will, according to the principles above, be given by the formula

To minimize this error, we must set

These values cannot be evaluated until the exact ratios $$\tfrac{n}{m},$$ $\tfrac{n'}{m'},$ etc. are known. In the absence of exact knowledge, it is safest to assume them equal to each other (see art. 11), in which case

i.e. the coefficients $$\alpha',$$ $\alpha'',$ etc., should be assumed equal to the relative weights of the various observations, taking the weight of the one corresponding to the error $$x$$ as the unit. With this assumption, let $$\sigma$$ denote, as above, the number of proposed errors. Then the average value of the expression

will be $$=m^2,$$ and when we take, for the true value of $$m^2,$$ the randomly determined value of this expression, the mean error to be feared will be

and, finally, if we are allowed to assume that the quantities $$n,$$ $$n',$$ $n,$ etc., are proportional to $$m,$$ $$m',$$ $m,$ etc., this expression reduces to

which is identical to what we found in the case where all observations were of the same type.

17.
When the value of a quantity, which depends on an unknown magnitude, is determined by an observation whose precision is not absolute, the result of this observation may provide an erroneous value for the unknown, but there is no room for discretion in this determination. But if several functions of the same unknown have been found by imperfect observations, we can obtain the value of the unknown either by any one of these observations, or by a combination of several observations, which can be carried out in infinitely many ways. The result will be subject, in all cases, to a possible error, and depending on the combination chosen, the mean error to be feared may be greater or smaller. The same applies if several observed quantities depend on multiple unknowns. Depending on whether the number of observations equals the number of unknowns, or is smaller or larger than this number, the problem will be determined, undetermined, or more than determined (at least in general), and in this third case, the observations can be combined in infinitely many ways to provide values for the unknowns. Among these combinations, the most advantageous ones must be chosen, i.e., those that provide values for which the mean error to be feared is as small as possible. This problem is certainly the most important one presented by the application of mathematics to natural philosophy.

In Theoria motus corporum coelestium we have shown how to find the most probable values of unknowns when the probability law of the observational errors is known, and since, in almost all cases, this law remains hypothetical by its nature, we have applied this theory to the highly plausible hypothesis that the probability of error $$x$$ is proportional to $$e^{-h^2 x^2}.$$ Hence this method that I have followed, especially in astronomical calculations, and which most calculators now use under the name of Method of Least Squares.

later considered the question from another point of view, and showed that this principle is preferable to all others, regardless of the probability law of the errors, provided that the number of observations is very large. But when this number is limited, the question remains open; so that, if we reject our hypothetical law, the method of least squares would be preferable to others, for the sole reason that it leads to simpler calculations.

We therefore hope to please geometers by demonstrating in this Memoir that the method of least squares provides the most advantageous combination of observations, not only approximately, but also absolutely, regardless of the probability law of errors and regardless of the number of observations, provided that we adopt for the mean error, not Laplace's definition, but the one which we have given in arts. 5 and 6.

It is necessary to warn here that in the following investigations, only random errors reduced by their constant part will be considered. It is up to the observer to carefully eliminate the causes of constant errors. We reserve for another occasion the examination of the case where observations are affected by an unknown constant error, and we will address this issue in another Memoir.

18.
Let $${U}$$ be a given function of the unknowns $${V},$$ $${V'},$$ ${V},$ etc.; we ask for the mean error $${M}$$ to be feared in determining the value of $${U}$$ when, instead of the true values of $${V},$$ $${V'},$$ ${V},$ etc., we take the values derived from independent observations; $$m,$$ $$m',$$ $m'',$ etc., being the mean errors corresponding to these various observations.

Solution. Let $$e,$$ $$e',$$ $e,$ etc. denote the errors of the observed values $${V},$$ $${V'},$$ ${V},$ etc.; the resulting error for the value of the function $${U}$$ can be expressed by the linear function

where $$\lambda,$$ $$\lambda',$$ $\lambda,$ etc., represent the derivatives $$\frac,$$ $$\frac,$$ $\frac,$ etc., when $${V},$$ $${V'},$$ ${V},$ etc., are replaced by their true values.

This value of $${E}$$ is evident if we assume the observations to be accurate enough so that the squares and products of the errors are negligible. It follows that the average value of $${E}$$ is zero, since we assume that the errors of the observations have no constant part. Now the mean error $${M}$$ to be feared in the value of $${U}$$ will be the square root of the average value of $${E}^2,$$ or equivalently $${M}^2$$ will be the average value of the sum

but the average value of $$\lambda^2 e^2$$ is $$\lambda^2 m^2,$$ that of $${\lambda'}^2 {e'}^2$$ is ${\lambda'}^2 {m'}^2$, etc., and finally the average values of the products $$2 \lambda \lambda' e e'$$ are all zero. Hence we find that

It is good to add several remarks to this solution.

I. Since we neglect powers of errors higher than the first, we can, in our formula, take for $$\lambda,$$ $$\lambda',$$ $\lambda,$ etc., the values of the differential coefficients $\frac,$ etc., derived from the observed values $${V},$$ $${V}',$$ ${V},$ etc. Whenever $${U}$$ is a linear function, this substitution is rigorously exact.

II. If instead of mean errors, one prefers to introduce weights $$p,$$ $$p',$$ $p'',$ etc. for the respective observations, with the unit being arbitrary, and $${P}$$ being the weight of the value of $${U}.$$  Then we will have

III. Let $${T}$$ be another function of $${V},$$ $${V}',$$ ${V}'',$ etc., and let

The error in the determination of $${T},$$ from the observed values $${V},$$ $${V}',$$ ${V}'',$ etc., will be

and the mean error to be feared in this determination will be

It is obvious that the errors $${E}$$ and $${E'}$$ will not be independent of each other, and the mean value of the product $${EE}'$$ will not be $$=0$$ like the mean value of $$ee',$$ but instead it will be equal to

IV. The problem includes the case where the values of the quantities $${V},$$ $${V}',$$ ${V}'',$ etc., are not immediately given by observation, but are deduced from any combinations of direct observations. For this extension to be legitimate, the determinations of these quantities must be independent, i.e., they must be provided by different observations. If this condition of independence is not fulfilled, the formula giving the value of $${M}$$ would no longer be accurate. For example, if the same observation were used both in determining $${V}$$ and in determining $${V}',$$ the errors $$e$$ and $$e'$$ would no longer be independent, and the mean value of the product $$ee'$$ would no longer be zero. If, in this case, the relationship between $${V}$$ and $${V'}$$ and the results of the simple observations from which they derive is known, we can calculate the mean value of the product $$ee',$$ as indicated in remark III, and consequently correct the formula which gives $${M}.$$

19.
Let $${V},$$ $${V'},$$ ${V''},$ etc., be functions of the unknowns $$x,$$ $$y,$$ $z,$ etc. Let $$\pi$$ be the number of these functions, and let $$\rho$$ be the number of unknowns. Suppose that observations have given, immediately or indirectly, $$V={L},$$ $$V'={L'},$$ $V={L},$ etc., and that these determinations are absolutely independent of each other. If $$\rho$$ is greater than $$\pi,$$ then the determination of the unknowns is an indeterminate problem. If $$\rho$$ is equal to $$\pi,$$ then each of the unknowns $$x,$$ $$y,$$ $z,$ etc., can be reduced to a function of $${V},$$ $${V'},$$ ${V''},$ etc., so that the values of the former can be deduced from the observed values of the latter, and the previous article will allow us to calculate the relative accuracy of these various determinations. If $$\rho$$ is less than $$\pi,$$ then each unknown $$x,$$ $$y,$$ $z,$ etc., can be expressed in infinitely many ways as a function of $${V},$$ $${V'},$$ ${V''},$ etc., and, in general, these values will be different; they should coincide if the observations were, contrary to our assumptions, rigorously accurate. It is clear, moreover, that the various combinations will provide results whose accuracy will generally be different.

Moreover, if, in the second and third cases, the quantities $${V},$$ $${V'},$$ ${V},$ etc., are such that $$\pi - \rho + 1$$ of them, or more, can be regarded as functions of the others, the problem is more than determined relative to these latter functions and indeterminate relative to the unknowns $$x,$$ $$y,$$ $z,$ etc.; and we could not even determine these latter unknowns, even if the functions $${V},$$ $${V'},$$ ${V},$ etc., were exactly known: but we exclude this case from our investigations.

If $${V},$$ $${V'},$$ ${V''},$ etc., are not linear functions of the unknowns, we can always assign them this form, by replacing the primitive unknowns with their difference from their approximate values, which we assume known; the mean errors to be feared in the determinations

being respectively denoted by $$m,$$ $$m',$$ $m,$ etc., and the weights of these determinations by $$p,$$ $$p',$$ $p,$ etc., so that

We will assume that both the ratios of the mean errors and the weights are known, one of which will be arbitrarily chosen. Finally, if we set

then things will proceed as if immediate observations, equally precise and with mean error $$ m \textstyle \sqrt{p^{\phantom{\prime}}},$$ had given

20.
Let $$v,$$ $$v',$$ $v,$ etc., be the following linear functions of the unknowns $$x,$$ $$y,$$ $z,$ etc.,''

{{MathForm1|(1)|$$\left\{\begin{array}{l}\begin{alignat}{6} &v  &{}={} & ax   &{}+{} & by   &{}+{} & cz   &{}+{} & \ldots &{}+{} & l,     \\ &v' &{}={} & a'x  &{}+{} & b'y  &{}+{} & c'z  &{}+{} & \ldots &{}+{} & l',    \\ &v &{}={} & ax &{}+{} & by &{}+{} & cz &{}+{} & \ldots &{}+{} & l'', \end{alignat}\\ \;\cdots\cdots\cdots\cdots\cdots\cdots\cdots \cdots \cdots\cdots\cdot \end{array}\right.$$}}

Among all systems of coefficients $$\varkappa,$$ $$\varkappa',$$ $\varkappa,$ etc., that identically satisfy

$$k$$ being independent of $$x,$$ $$y,$$ $z,$ etc., find the one for which $$\varkappa^2 + {\varkappa'}^2 + {\varkappa}^2+\ldots$$ obtains its minimum value.''

Solution. — Let us set

{{MathForm1|(2)|$$\left\{ \begin{array}{l}\begin{alignat}{4} a v  &{}+{}& a' v' &{}+{}& a v &{}+{}& \ldots &= \xi, \\ b v  &{}+{}& b' v' &{}+{}& b v &{}+{}& \ldots &= \eta, \\ c v  &{}+{}& c' v' &{}+{}& c v &{}+{}& \ldots &= \zeta, \end{alignat} \\ \;\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdot\cdot \end{array} \right. $$|m=1em}} $$\xi,$$ $$\eta,$$ $$\zeta$$ are linear functions of $$x,$$ $$y,$$ $$z,$$ and we have {{MathForm1|(3)|$$\left\{ \begin{array}{l}\begin{alignat}{5} \xi  &= x \Sigma a^2 &{}+{}& y \Sigma ab  &{}+{}& z \Sigma ac   &{}+{}& \ldots &{}+{}& \Sigma al, \\[0.75ex] \eta &= x \Sigma ab  &{}+{}& y \Sigma b^2 &{}+{}& z \Sigma bc   &{}+{}& \ldots &{}+{}& \Sigma bl, \\[0.75ex] \zeta &= x \Sigma ac &{}+{}& y \Sigma bc  &{}+{}& z \Sigma c^2  &{}+{}& \ldots &{}+{}& \Sigma cl, \end{alignat} \\ \;\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdot \end{array} \right.$$|m=1em}} where $$\Sigma a^2$$ denotes the sum $$a^2 + {a'}^2 + {a''}^2 + \ldots,$$ and similarly for the other sums.

The number of quantities $$\xi,$$ $$\eta,$$ $\zeta,$ etc., is equal to the number of unknowns $$x,$$ $$y,$$ $z,$ etc., namely $$\rho$$. Thus, by elimination, one can obtain an equation of the following form,

which will be identically satisfied if we replace $$\xi,$$ $$\eta,$$ $$\zeta$$ with their values from (3). Consequently, if we set

{{MathForm1|(4)|$$\left\{\begin{alignat}{8} &a  &&(\alpha\alpha) &{}+{}& b   &&(\alpha\beta) &{}+{}& c   &&(\alpha\gamma) &{}+{}& \ldots &&= \alpha, \\ &a' &&(\alpha\alpha) &{}+{}& b'  &&(\alpha\beta) &{}+{}& c'  &&(\alpha\gamma) &{}+{}& \ldots &&= \alpha', \\ &a &&(\alpha\alpha) &{}+{}& b &&(\alpha\beta) &{}+{}& c &&(\alpha\gamma) &{}+{}& \ldots &&= \alpha, \\ \end{alignat}\right.$$|m=1em}}

then we will have identically

This equation shows that among the different systems of coefficients $$\varkappa,$$ $$\varkappa',$$ $\varkappa'',$ etc., we must consider the system

Moreover, for any system, we will have identically

and this equation, being identical, leads to the following:

Adding these equations after multiplying them, respectively, by $$(\alpha\alpha),$$ $$(\alpha\beta),$$ $(\alpha\gamma),$ etc., we will have, by virtue of the system (4),

which is the same as

thus, the sum

will have its minimum value when $$\varkappa = \alpha,$$ $$ \varkappa' = \alpha',$$ $\varkappa = \alpha,$ etc. Q.E.I.

Moreover, this minimum value will be obtained as follows. Equation (5) shows that we have

Let's multiply these equations, respectively, by $$(\alpha\alpha),$$ $$(\alpha\beta),$$ $(\alpha\gamma),$ etc., and add them; considering the relations (4), we find

21.
When the observations have provided approximate equations $$v=0,$$ $$v'=0,$$ $v''=0,$ etc., it will be necessary, to determine the unknown $$x,$$ to choose a combination of the form

such that the unknown $$x$$ acquires a coefficient equal to $$1$$, and that the other unknowns are eliminated.

According to art. 18, the weight of this determination will be given by

According to the previous article, the most suitable determination will be obtained by taking $$\varkappa = \alpha,$$ $$\varkappa' = \alpha', $$ $\varkappa = \alpha, $ etc. Then $$x$$ will have the value $${A},$$ and it is clear the same value would be obtained (without knowing the multipliers $$\alpha,$$ $$\alpha',$$ $\alpha'',$ etc.), by performing elimination on the equations $$\xi = 0,$$ $$\eta = 0,$$ $\zeta = 0,$ etc. The weight of this determination will be given $$=\tfrac{1}{(\alpha\alpha)},$$ and the mean error to be feared will be

A similar approach would lead to the most suitable values of the other unknowns $$y,$$ $z,$ etc., which would be those obtained by performing eliminating on the equations $$\xi = 0,$$ $$\eta = 0,$$ $$\zeta = 0,$$ etc.

If we denote the sum $$v^2 + { v'}^2 + { v''}^2 + \ldots,$$ or equivalently

by $$\Omega$$, then it is clear that $$2\xi,$$ $$2\eta,$$ $$2\zeta,$$ etc. will be the partial differential quotients of the function $$\Omega,$$ i.e.

Therefore, the values of the unknowns that are deduced from the most suitable combination, and which we can call the most plausible values, are precisely those that minimize $$\Omega$$. Now $${V-L}$$ represents the difference between the observed value and the computed value. Thus, the most plausible values of the unknowns are those that minimize the sum of the squares of the differences between the calculated and observed values of the quantities $${V},$$ $${V'},$$ ${V''},$ etc., these squares being respectively multiplied by the weight of the observations. I had established this principle a long time ago through other considerations, in Theoria Motus Corporum Coelestium.

If one wants to assign the relative precision of each determination, it is necessary to deduce the values of $$x,$$ $$y,$$ $z,$ etc. from the equations (3), which gives them in the following form:

{{MathForm1|(7)|$$\left\{\begin{alignat}{5} x &= {A} &{}+{}& (\alpha\alpha) &\, \xi &{}+{} (\alpha\beta) &\, \eta &{}+{} (\alpha\gamma) &\, \zeta + \ldots, \\ y &= {B} &{}+{}& (\beta\alpha) &\, \xi &{}+{} (\beta\beta)  &\, \eta &{}+{} (\beta\gamma)  &\, \zeta + \ldots, \\ z &= {C} &{}+{}& (\gamma\alpha) &\, \xi &{}+{} (\gamma\beta) &\, \eta &{}+{} (\gamma\gamma) &\, \zeta + \ldots. \end{alignat}\right.$$}}

Accordingly, the most plausible values of the unknowns $$x,$$ $$y,$$ $z,$ etc., will be $${A},$$ $${B},$$ ${C},$ etc. The weights of these determinations will be $$\tfrac{1}{(\alpha\alpha)},$$ $$\tfrac{1}{(\beta\beta)},$$ $\tfrac{1}{(\gamma\gamma)},$ etc. and the mean errors to be feared will be in agreement with the results obtained in Theoria Motus Corporum Coelestium.

22.
The case where there is only one unknown is the most frequent and simplest of all. In this case we have $${V} = x,$$ $${V'} = x,$$ $ {V} = x,$ etc. We will then have $$a = \textstyle\sqrt{p^{\phantom{\prime}}},$$ $$ a' = \textstyle\sqrt{p'},$$ $a = \textstyle\sqrt{p}$ etc., $$l = -{L}\sqrt{p^\stackrel{}{}},$$ $$ l' = -{L'}\sqrt{p'},$$ $ l = -{L}\sqrt{p}$ etc., and consequently,

Hence

Therefore, if by several observations that do not have the same precision and whose respective weights are p, p', p'', etc., we have found, for the same quantity, a first value {L}, a second {L'}, a third {L''}, etc., then the most plausible value will be

and the weight of this determination will be If all observations are equally plausible, then the most probable value will be

i.e. the arithmetic mean of the observed values; taking the weight of an individual observation as the unit, the weight of the average will be \pi.

23.
A number of investigations still remain to be discussed, through which the preceding theory will be clarified and extended.

Let us first investigate whether the elimination used to express the variables x, y, z, etc., in terms of \xi, \eta, \zeta, etc., is always possible. Since the number of equations is equal to the number of unknowns, we know that this elimination will be possible if \xi, \eta, \zeta, etc. are independent of each other; otherwise, it is impossible.

Suppose, for a moment, that \xi, \eta, \zeta, etc. are not independent, but rather there exists between these quantities an identical equation We will then have

Let us set

from which it follows that Multiplying the equations (1) resp. by \theta, \theta', \theta'', etc., and adding, we obtain and this equation leads to \theta = 0,  etc. From this we conclude, first of all, {K}=0. Secondly, the equations (1) show that the functions v, v', v'', etc., are such that their values do not change when the variables x, y, z, etc., increase or decrease proportionally to {F}, {G}, {H}, etc. respectively. It is clear that the same holds for the functions {V}, {V}', {V}, etc.: but this can only happen in the case where it would be impossible to determine x, y, z, etc. using the values of {V}, {V}', {V}, etc., even if these were exactly known; but then the problem would be indeterminate by its nature, and we will exclude this case from our investigations.

24.
If \beta, \beta', \beta, etc. denote multipliers playing the same role relative to the unknown y, as the multipliers \alpha, \alpha', \alpha, etc. relative to the unknown x, i.e. so that we have

then we will identically have

Let \gamma, \gamma', \gamma'', etc. be the analogous multipliers relative to the variable z, so that we have:

and consequently,

In the same way as we found in art. 20 that

we will find here

and so on.

We will also have, as in art. 20

If we multiply the values \alpha, \alpha', \alpha, etc. (art. 20. (4)), respectively, by \beta, \beta', \beta, etc., and add; we obtain

If we multiply \beta, \beta', \beta, etc., respectively, by \alpha, \alpha', \alpha, etc., and add, we will find

In the same manner, we find

25.
Let \lambda, \lambda', \lambda, etc. denote the values taken by the functions v, v', v, etc., when x, y, z, etc. are replaced by their most plausible values, {A}, {B}, {C}, etc., i.e.

If we set

so that {M} is the value of the function \Omega corresponding to the most plausible values of the variables, and therefore, as was shown in art. 20, the minimum value of \Omega. Then the value of will be \xi, corresponding to x = {A}, y = {B}, \end{aligned},$$ and this value is zero, according to the way $${A},$$ $${B},$$ ${C},$ etc. have been obtained. Thus, we have

and similarly we would obtain

and

Finally, multiplying the values of $$\lambda,$$ $$\lambda',$$ $\lambda,$ etc. respectively by $$\lambda,$$ $$\lambda',$$ $$\lambda,$$ and adding, we get $$l\lambda + l'\lambda' + l\lambda + \ldots = \lambda^2 + {\lambda'}^2 + {\lambda''}^2 + \ldots, $$ or

26.
Replacing $$x,$$ $$y,$$ $z,$ etc., with the expressions (7) from art. 21 in the equation $$v = ax + by + cz + \ldots + l,$$ we find, through the same reductions as before,

Multiplying either these equations or the equations (1) of art. 20, by $$\lambda,$$ $$\lambda',$$ $\lambda'',$ etc., and then adding, we obtain the identity

27.
The function $$\Omega$$ can take several forms, which are worth developing.

Let us square the equations (1) art. 20, and add them. Then we find

this is the first form.

Next let us multiply the same equations by $$v,$$ $$v',$$ $v'',$ etc. respectively, and add. Then we obtain $$\Omega = \xi x + \eta y + \zeta z + \ldots + l v + l' v' + l v + \ldots ;$$ and replacing $$v,$$ $$v',$$ $v'',$ etc., with the values indicated in the previous article, we find that $$\Omega = \xi x + \eta y + \zeta z + \ldots - {A} \xi - {B} \eta - {C} \zeta - \ldots + {M} ,$$ or $$\Omega = \xi\,(x - {A}) + \eta\,(y - {B}) + \zeta\,(z - {C}) + \ldots + {M} :$$ this is the second form.

Finally, replacing, in this second form, $$x - {A},$$ $$y - {B},$$ $z - {C},$ etc. by the expressions (7) art. 21, we obtain the 'third form':

We can also give a fourth form which results automatically from the third form and the formulas of the previous article:

From this last form we clearly see that $${M}$$ is the minimum value of $$\Omega.$$

28.
Let $$e,$$ $$e',$$ $e'',$ etc., be the errors made in the observations that gave $${V}  = {L}, $$ $${V}'  = {L}',$$ ${V} = {L}, $ etc. Then the true values of the functions $${V},$$ $${V}',$$ ${V},$ etc., will be $${L}-e,$$ $$ {L}'-e',$$ $ {L}-e,$ etc. respectively, and the true values of $$v,$$ $$v',$$ $v,$ etc., will be $$-e\sqrt{p^{\phantom{\prime}}},$$ $$ -e'\sqrt{p'},$$ $$ -e\sqrt{p},$$ etc. respectively. therefore, the true value of $$x$$ will be

and the error made in the most suitable determination of the unknown $$e,$$ which we will denote by $${E}x,$$ will be

Similarly, the error made in the most suitable determination of the value of $$y$$ will be

The average value of the square $${({E}x)}^2$$ will be

The average value of $${({E}y)}^2$$ will similarly be $$= m^2 p \, (\beta\beta),$$ as shown above. We can also determine the average value of the product $${E}x \,.\, {E}y,$$ which will be

These results can be stated more briefly as follows:

The average values of the squares $${({E}x)}^2,$$ ${({E}y)}^2,$ etc., are respectively equal to the products of $$\tfrac{1}{2} m^2p$$ with the second-order partial differential quotients

and the average value of a product such as $${E}x.{E}y$$ is the product of $$\tfrac{1}{2}m^2p$$ with $$\frac{{d}^2\Omega}{{d}\xi\,{d}\eta},$$ where $$\Omega$$ is regarded as a function of $$\xi,$$ $$\eta,$$ $\zeta,$ etc.}

29.
Let $$t$$ be a given linear function of the quantities $$x,$$ $$y,$$ $z,$ etc., i.e.

the value of $$t$$ deduced from the most plausible values of $$x,$$ $$y,$$ $z,$ etc., will then be $$= f\,{A} + g\,{B} + h\,{C} + \ldots + k,$$ and we denote this by $${K}.$$ The error thus committed will be

which we denote by $${E}t.$$ The average value of this error will obviously be zero, meaning the error will not contain a constant part, but the average value of $${({E}t)}^2,$$ i.e., the sum

will, according to the preceding article, be equal to the product of $$m^2p$$ with the sum

i.e., the product of $$m^2p$$ with the value produced by the function $$\Omega - {M}$$ when we substitute $$\begin{aligned}\xi &= f, & \eta &= g, & \zeta &= h, \ldots. \end{aligned}$$

If we let $$\omega$$ denote this value of $$\Omega - {M},$$ then the mean error to be feared when we take $$t = {K}$$ will be $$m\sqrt{p^{\phantom{\prime}}}$$ and the weight of this determination will be $$\tfrac{1}{\omega}$$.

Since we have identically

$$\omega$$ will be equal to the value of the expression $$(x - {A})\,f + (y-{B})\,g + (z - {C})\,h + \ldots$$ or the value produced by $$(t = {K})$$ when we substitute for $$x,$$ $$y,$$ $z,$ etc. the values corresponding to $$\xi = f,$$ $$\eta = g,$$ $\zeta = h,$ etc..

Finally, observing that $$t,$$ expressed as a function of the quantities $$\xi,$$ $$\eta,$$ $\zeta,$ etc., will have $${K}$$ as its constant part, if we suppose that

then we will have

30.
We have seen that the function $$\Omega$$ attains its absolute minimum $${M},$$ when we substitute $$x = {A},$$ $$ y = {B},$$ $ z = {C},$ etc. or, equivalently, $$\xi = 0,$$ $$ \eta = 0,$$ $ \zeta = 0,$ etc. If we assign another value to one of the unknowns, e.g. $$ x = {A} + \Delta,$$ while the other unknowns remain variable, $$\Omega$$ may acquire a relative minimum value, which can be obtained from the equations

Therefore, we must have $$\eta = 0,$$ $ \zeta = 0,$etc., and since

we have

Likewise, we have

and the relative minimum value of $$\Omega$$ will be

Reciprocally, we conclude that if $$\Omega$$ is not to exceed $${M} + \mu^2,$$ then the value of $$x$$ must necessarily be between the limits $${A} - \mu \sqrt{(\alpha\alpha)}$$ and $${A} + \mu \sqrt{(\alpha\alpha)}.$$ It is important to note that $$\mu \sqrt{(\alpha\alpha)}$$ becomes equal to the mean error to be feared in the most plausible value of $$x,$$ if we set $$\mu = m \sqrt{p^{\phantom{\prime}}} ;$$ i.e., if $$\mu$$ is the mean error of observations whose weights are $$=1$$.

More generally, let us find the smallest value of the function $$\Omega$$ that can correspond to a given value of $$t,$$ where $$t$$ denotes, as in the previous article, a linear expression $$fx + gy + hz + \ldots + k$$ whose most plausible value is $${K}$$. Let us denote by the prescribed value of $$t$$ by $${K} + \kappa.$$ According to the theory of maxima and minima, the solution to the problem will be given by the equations

or $$\xi  = \theta\,f,$$ $$\eta  = \theta\,g,$$ $$ \zeta = \theta\,h,$$ etc., where $$\theta$$ denotes an as yet undetermined multiplier. If, as in the previous article, we identically set,

then we will have

or

where $$\omega$$ has the same meaning as in the previous article.

Since $$\Omega - {M}$$ is a homogeneous function of the second degree with respect to the variables $$\xi,$$ $$\eta,$$ $$\zeta,$$ etc., its value when $$\xi = \theta\,f,$$ $$ \eta = \theta\,g,$$ $$ \zeta = \theta\,h,$$ etc. will evidently be $$=\theta^2\omega ,$$ and thus the minimum value of $$\Omega,$$ when $$t = {K} + \kappa,$$ will be $$={M} + \theta^2 \omega = {M} + \tfrac{\kappa^2}{\omega}.$$ Reciprocally, if $$\Omega$$ must remain less than a given value $${M} + \mu^2,$$ the value of $$t$$ will necessarily be between the limits $${K} - \mu\sqrt{\omega^\stackrel{}{}},$$ $${K} + \mu\sqrt{\omega^\stackrel{}{}},$$ and $$\mu\sqrt{\omega^\stackrel{}{}}$$ will be the mean error to be feared in the most plausible value of $$t,$$ if $$\mu$$ represents the mean error of observations whose weights are $$=1$$.

31.
When the number of unknowns $$x,$$ $$y,$$ $z,$ etc. is quite large, the determination of the numerical values of $${A},$$ $${B},$$ $${C},$$ etc. by ordinary elimination is quite tedious. For this reason we have indicated, in Theoria Motus Corporum Coelestium art. 182, and later developed, in Disquisitione de elementis ellipticis Palladis (Comm. recent. Soc. Gotting Vol. I), a method that simplifies this work as much as possible. Namely, the function $$\Omega$$ must be reduced to the following form:

where the divisors $$\mathfrak{A}^0,$$ $$\mathfrak{B}',$$ $$\mathfrak{C},$$ $$\mathfrak{D},$$ etc., are determined quantities; $$u^0,$$ $$u',$$ $$u,$$ etc., are linear functions of $$x,$$ $$y,$$ $$z,$$ etc., such that the second $$u'$$ does not contain $$x,$$ the third $$u$$ contains neither $$x$$ nor $$y,$$ the fourth contains neither $$x,$$ nor $$y,$$ nor $$z,$$ and so on, so that the last $$u^{(\pi - 1)}$$ contains only the last of the unknowns $$x,$$ $$y,$$ $$z,$$ etc.; and finally, the coefficients of $$x,$$ $$y,$$ $$z,$$ etc., in $$u^0,$$ $$u',$$ $$u,$$ etc., are respectively equal to $$\mathfrak{A}^0,$$ $$\mathfrak{B}',$$ $$\mathfrak{C},$$ etc. Then we set $$u^0 = 0,$$ $$ u' = 0,$$ $$ u = 0, $$ $$ u = 0, $$ etc. and we will easily obtain the values of $$x,$$ $$y,$$ $z,$ etc. by solving these equations, starting with the last one. I do not believe it necessary to repeat the algorithm that leads to the transformation of the function $$\Omega$$.

However, the elimination required to find the weights of these determinations requires even longer calculations. We have shown in the Theoria Motus Corporum Coelestium that the weight of the last unknown, (which appears by itself in $$u^{(\pi - 1)}),$$ is equal to the last term in the series of divisors $$\mathfrak{A}^0,$$ $$\mathfrak{B}',$$ $$\mathfrak{C}'',$$ etc. This is easily found; hence, several calculators, wanting to avoid cumbersome elimination, have had the idea, in the absence of another method, to repeat the indicated transformation by successively considering each unknown as the last one. Therefore, I hope that geometers will appreciate my indication of a new method for calculating the weights of determinations, which seems to leave nothing more to be desired on this point.

Setting {{MathForm1|(1)|$$\left\{\begin{array}{l}\begin{alignat}{6} &u^0 &{}={}& \mathfrak{A}^0 x &{}+{}& \mathfrak{B}^0 y &{}+{}& \mathfrak{C}^0 z &{}+{}& \ldots &{}+{}& \mathfrak{L}^0, \\ &u' &{}={}& & & \mathfrak{B}'\,y &{}+{}& \mathfrak{C}'\,z &{}+{}& \ldots &{}+{}& \mathfrak{L}'\,, \\ &u &{}={}& & & & & \mathfrak{C} z &{}+{}& \ldots &{}+{}& \mathfrak{L}'', \end{alignat}\\ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \end{array}\right.$$}} we have identically

and from this we deduce: {{MathForm1|(2)|$$\left\{\begin{array}{l}\begin{alignat}{3} &\xi  &{}={}& u^0, \\[0.25em] &\eta &{}={}& \frac{\mathfrak{B}^0}{\mathfrak{A}^0} u^0 &{}+{}& u',  \\ &\zeta &{}={}& \frac{\mathfrak{C}^0}{\mathfrak{A}^0} u^0 &{}+{}& \frac{\mathfrak{C}'}{\mathfrak{B}'} u' &{}+{}& u'', \end{alignat}\\ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \end{array}\right.$$}}

The values of $$u^0,$$ $$u',$$ $u'',$ etc. deduced from these equations will be presented in the following form:

{{MathForm1|(3)|$$\left\{\begin{array}{l}\begin{alignat}{3} &u^0 &{}={}& \xi, \\ &u' &{}={}& {A}' \xi &{}+{}& \eta,  \\ &u &{}={}& {A} \xi &{}+{}& {B}'' \eta &{}+{}& \zeta, \end{alignat} \\ \cdots\cdots\cdots\cdots\cdots\cdots\cdots \end{array}\right.$$}} By taking the complete differential of the equation

we obtain

and thus

This expression must be equivaleny to the one obtained from the equations (3),

and therefore we have {{MathForm1|(4)|$$\left\{\begin{array}{l} \begin{alignat}{6} x &{}={}& \frac{u^0}{\mathfrak{A}^0} &{}+{}& {A}'\,\frac{u'}{\mathfrak{B}'} &{}+{}& {A}\,\frac{u}{\mathfrak{C}''} &{}+{}& \ldots &{}+{}& {A}, \\ y &{}={}& & & \frac{u'}{\mathfrak{B}'} &{}+{}& {B}\,\frac{u}{\mathfrak{C}''} &{}+{}& \ldots &{}+{}& {B}, \\ z &{}={}& & & & & \frac{u}{\mathfrak{C}} &{}+{}& \ldots &{}+{}& {C}, \end{alignat}\\ \,\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \end{array}\right.$$}} By substituting in these expressions the values of $$u^0,$$ $$u',$$ and $u'',$ etc. obtained from the equations (3), we will have performed the elimination. For the determination of the weights, we have

{{MathForm1|(5)|$$\left\{\begin{array}{l} \begin{alignat}{6} &(\alpha\alpha) &{}={}& \frac{1}{\mathfrak{A}^0} &{}+{}& \frac{{{A}'}^2}{\mathfrak{B}'} &{}+{}& \frac{{{A}}^2}{\mathfrak{C}} &{}+{}& \frac{{{A}}^2}{\mathfrak{D}} &{}+{}& \ldots,\\ &(\beta\beta) &{}={}& & & \frac{1}{\mathfrak{B}'} &{}+{}& \frac{{{B}}^2}{\mathfrak{C}} &{}+{}& \frac{{{B}}^2}{\mathfrak{D}} &{}+{}& \ldots,\\ &(\gamma\gamma) &{}={}& & & & & \frac{1}{\mathfrak{C}''} &{}+{}& \frac{ {C}^2}{\mathfrak{D}} &{}+{}& \ldots.\end{alignat}\\ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdot \end{array}\right.$$}}

The simplicity of these formulas leaves nothing to be desired. Equally simple formulas could be found to express the other coefficients $$(\alpha\beta),$$ $$(\alpha\gamma),$$ and $(\beta\gamma),$ etc.; however, as their use is less frequent, we will refrain from presenting them.

33.
The importance of the subject has prompted us to prepare everything for the calculation and to form explicit expressions for the coefficients $${A}',$$ $${A},$$ ${A}$ etc.,$${B},$$ ${B},$ etc. etc. This calculation can be approached in two ways. The first involves substituting the values of $$u^0,$$ $$u',$$ and so forth, deduced from the equations (3) into the equations (2), and the second involves substituting the values $$\xi,$$ $$\eta,$$ $$\zeta,$$ from the equations (2) into the equations (3). The first method leads to the following formulas:

These formulas will determine $${A}',$$ $${A}'',$$ and so on.

We will then have,

which will determine $${B}'',$$ and so forth; then

which will determine $${C}'''$$ etc., and so on.

The second method yields the following system:

from which we deduce $${A}',$$

from which we deduce $${B}$$ and $${A},$$

from which we deduce $${C},$$ $${B},$$ $${A}''',$$ and so on.

Both systems of formulas offer nearly equal advantages when seeking the weights of the determinations of all unknowns $$x,$$ $$y,$$ and so forth; however, if only one of the quantities $$(\alpha\alpha),$$ $$(\beta\beta),$$ and so forth is required, the first system is much preferable.

Moreover, the combination of equations (1) and (4) yields the same formulas, and provides, in addition, a second way to obtain the most plausible values $${A},$$ $${B},$$ and so forth, which are

The other calculation is identical to the ordinary calculation in which it is assumed $$u^0 = 0,$$ $$ u' = 0,$$ $ u''= 0,$ etc.

34.
The results obtained in art. 32 are only particular cases of a more general theorem which can be stated as follows:

If $$t$$ represents the following linear function of the unknowns $$x,$$ $$y,$$ $z,$ etc.,

whose expression in terms of the variables $$u^0,$$ $$u',$$ $u,$ etc., is''

then $${K}$$ will be the most plausible value of $$t,$$ and the weight of this determination will be

Proof. The first part of the theorem is obvious, since the most plausible value of $$t$$ must correspond to the values $$u^0 = 0,$$ $$ u' = 0,$$ $ u''= 0,$ etc.

To demonstrate the second part, let's note that we have

and consequently, when

we have

whatever the differentials $${d}x,$$ $${d}y,$$ ${d}z,$ etc. Hence, assuming always, $$\begin{aligned}\xi &= f, & \eta &= g, & \zeta &= h, \ldots, \end{aligned}$$ we obtain

Now it is easily seen that if the differentials $${d}x,$$ $${d}y,$$ ${d}z,$ etc. are independent of each other, so will be $${d}u^0,$$ $${d}u',$$ ${d}u'',$ etc., therefore, we will have,

Hence, the value of $$\Omega$$ corresponding to the same assumptions, will be

which, by art. 29, demonstrates the truth of our theorem.

Moreover, if we wish to perform the transformation of the function $$t$$ without resorting to formulas (4) of art. 32, we immediately have the relations

which will allow us to determine $$k^0,$$ $$k',$$ $k'',$ etc., and we will finally have

35.
We will particularly address the following problem, both because of its practical utility and the simplicity of the solution:

Find the changes that the most plausible values of the unknowns undergo by adding a new equation, and assign the weights of these new determinations.

Let us keep the previous notations. The primitive equations, reduced to have a weight of unity, will be $$\begin{aligned}v &= 0, & v' &= 0, & v'' &= 0, \ldots ; \end{aligned}$$ we will have $$\Omega = { v}^2 + { v'}^2 + { v''}^2 + \ldots,$$ and $$\xi,$$ $$\eta,$$ $\zeta,$ etc., will be the partial derivatives

Finally, by elimination, we will have {{MathForm1|(1)|$$\left\{\begin{array}{l} \begin{alignat}{5} x &{}={} {A} &&{}+{}& (\alpha\alpha) & \,\xi {}+{}& (\alpha\beta) & \,\eta {}+{}& (\alpha\gamma) & \,\zeta + \ldots, \\ y &{}={} {B} &&{}+{}& (\alpha\beta) & \,\xi {}+{}& (\beta\beta) & \,\eta {}+{}&(\beta\gamma) & \,\zeta + \ldots, \\ z &{}={} {C} &&{}+{}& (\alpha\gamma) & \,\xi {}+{}& (\beta\gamma) & \,\eta {}+{}& (\gamma\gamma) & \,\zeta + \ldots, \end{alignat}\\ \;\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdot\cdot \end{array}\right.$$}}

Now suppose we have a new approximate equation $$v^\ast = 0$$ (which we assume to have a weight equal to unity), and we seek the changes undergone by the most plausible values of $${A},$$ $${B},$$ ${C},$ etc., and of the coefficients $$(\alpha\alpha),$$ $(\beta\beta),$ etc..

Let us set

and let

be the result of the elimination. Finally, let

which, taking into account the equations (1), becomes

and let

It is clear that $${K}$$ will be the most plausible value of the function $$v^\ast,$$ as resulting from the primitive equations, without considering the value $$0,$$ provided by the new observation, and $$\tfrac{1}{\omega}$$ will be the weight of this determination.

Now we have

and consequently,

or

Furthermore,

From this, we deduce,

which will be the most plausible value of $$x,$$ deduced from all observations.

We will also have

thus

will be the weight of this determination.

Similarly, for the most plausible value of $$y,$$ deduced from all observations, we find

the weight of this determination will be

and so on. Q.E.I.

Let us add some remarks.

I. After substituting the new values $${A}^\ast,$$ $${B}^\ast,$$ ${C}^\ast,$ etc., the function $$v^\ast$$ will obtain the most plausible value

and since we have, identically,

the weight of this determination, according to art. 29, will be

These results could be deduced immediately from the rules explained at the end of art. 21. The original equations had, indeed, provided the determination $$v^\ast={K},$$ whose weight was $$\tfrac{1}{\omega}.$$ A new observation gives another determination $$v^\ast = 0,$$ independent of the first, whose weight is $$=1,$$ and their combination produces the determination $$v^\ast = \tfrac{1+\omega}$$ with a weight of $$\tfrac{1}{\omega} + 1.$$

II. It follows from the above that, for $$x = {A}^\ast,$$ $$ y = {B}^\ast, $$ $ z = {C}^\ast$ etc. we must have $$\xi^\ast = 0,$$ $$ \eta^\ast = 0,$$ $ \zeta^\ast= 0$ etc., and consequently,

Furthermore, since

we must have

and

III. Comparing these results with those of the art. 30, we see that here the function $$\Omega$$ has the smallest value it can obtain when subjected to the condition $$v^\ast = \tfrac{1+\omega}.$$

36.
We will give here the solution to the following problem, which is analogous to the previous one, but we will refrain from indicating the demonstration, which can be easily found, as in the previous article.

Find the changes in the most plausible values of the unknowns and the weights of the new determinations when changing the weight of one of the primitive observations.

Suppose that after completing the calculation, it is noticed that the weight which has been assigned to an observation is too strong or too weak, e.g. the first one which gave $${V} = {L},$$ and that it would be more accurate to assign it the weight $$p^\ast,$$ instead of the weight $$p.$$ It is not necessary to then restart the calculation. Instead it is convenient to form the corrections using the following formulas.

The most plausible values of the unknowns will be corrected as follows:

and the weights of these determinations will be found upon dividing unity by

respectively.

This solution applies in the case where, after completing the calculation, it is necessary to completely reject one of the observations, since this amounts to making $$p^\ast = 0$$; similarly, $$p^\ast = \infty$$ will be suitable for the case where the equation $${V} = {L},$$ which in the calculation had been regarded as approximate, is in fact absolutely precise.

If, after completing the calculation, several new equations were to be added to those proposed, or if the weights assigned to several of them were incorrect, the calculation of the corrections becomes too complicated, and it is preferable to start over.

37.
In the arts. 15 and 16, we have given a method to approximate the accuracy of a system of observations; but this method assumes that the real errors encountered in a large number of observations are known exactly; however, this condition is rarely fulfilled, if ever.

If the quantities for which the observation provides approximate values depend on one or more unknowns, according to a given law, then the method of least squares allows us to find the most plausible values of these unknowns. If we then calculate the corresponding values of the observed quantities, they can be regarded as differing little from the true values, so that their differences with the observed values will represent the errors committed, with a certainty that will increase with the number observations. This is the procedure followed in practice by calculators, who have attempted, in complicated cases, to retrospectively evaluate the precision of the observations. Although sufficient in many cases, this method is theoretically inaccurate and can sometimes lead to serious errors; therefore, it is very important to treat the issue with more care.

In the following discussion, we retain the notation used in art. 19. The method in question consists of considering $${A},$$ $${B},$$ ${C},$ etc., as the true values of the unknowns $$x,$$ $$y,$$ $z,$ etc., and $$\lambda,$$ $$\lambda',$$ $\lambda,$ etc., as those of the functions $$v,$$ $$v',$$ $v,$ etc. If all observations have equal precision and their common weight $$p = p' = p'' = \ldots$$ is taken to be unity, these same quantities, changed in sign, represent, under this assumption, the errors of the observations. Consequently, according to art. 15,

will be the mean error of the observations. If the observations do not have the same precision, then $$-\lambda,$$ $$-\lambda',$$ $-\lambda'',$ etc., represent the errors of the observations, respectively multiplied by the square roots of the weights, and the rules of art. 16 lead to the same formula,

which already expresses the mean error of these observations, when their weight is $$=1$$. However, it is clear that an exact calculation would require replacing $$\lambda,$$ $$\lambda',$$ $\lambda,$ etc. with the values of $$v,$$ $$v',$$ $v,$ etc., deduced from the true values of the unknowns $$x,$$ $$y,$$ $z,$ etc., and replacing the quantity $${M}$$ by the corresponding value of $$\Omega.$$ Although we cannot assign this latter value, we are nonetheless certain that it is greater than $${M}$$ (which is its minimum possible value), and it would only reach this limit in the infinitely unlikely case where the true values of the unknowns coincide with the most plausible ones. We can therefore affirm, in general, that the mean error calculated by ordinary practice is smaller than the exact mean error, and consequently, that too much precision is attributed to the observations. Now let us see what a rigorous theory yields.

38.
First of all, we need to determine how the quantity $${M}$$ depends on the true errors of the observations. As in art. 28, Let us denote these errors by $$e,$$ $$e',$$ $e'',$ etc., and let us set, for simplicity,

and

Let $${A} - x^0,$$ $${B} - y^0,$$ ${C}- z^0,$ etc., be the true values of the unknowns $$x,$$ $$y,$$ $z,$ etc., for which $$\xi,$$ $$\eta,$$ $\zeta,$ etc., are, respectively, $$-\xi^0,$$ $$-\eta^0,$$ $-\zeta^0,$ etc. The corresponding values of $$v,$$ $$v',$$ $v',$ etc., will obviously be $$\begin{aligned}- \varepsilon&, & - \varepsilon'&, & - \varepsilon'', \ldots ;& \end{aligned}$$ so that we will have

Finally,

will be the value of the function $$\Omega,$$ corresponding to the true values of the $$x,$$ $$y,$$ $z,$ etc. Since we also have identically

we will also have

From this, it is clear that $${M}$$ is a homogeneous function of the second degree of the errors $$e,$$ $$e',$$ $e'',$ etc.; for various values of the errors this function may become greater or smaller. However, the extent of the errors remains unknown to us, so it is good to carefully examine the function $${M}$$, and to first calculate its average value according to the elementary calculus of probability. We will obtain this average value by replacing the squares $$e^2,$$ ${e'}^2,$ etc. with $$m^2,$$ ${m'}^2,$ etc., and omitting the terms in $$ee',$$ $ee,$ etc., whose average value is zero; or equivalently, by replacing each square $$\varepsilon^2,$$ $${\varepsilon'}^2,$$ ${\varepsilon}^2,$ $\ldots$, by $$\mu^2,$$ and neglecting $$\varepsilon\varepsilon',$$ $\varepsilon\varepsilon'',$ $\ldots$. Accordingly, the term $$\Omega^0$$ will provide $$\pi\mu^2$$; the term $$-x^0\xi^0$$ will produce

each of the other terms will also give $$-\mu^2,$$ so that the total average value will be $$=(\pi - \rho) \, \mu^2,$$ where $$\pi$$ denotes the number of observations, and $$\rho$$ denotes the number of unknowns. Due to errors offered by chance, the true value of $${M}$$ may be greater or smaller than this average value, but the difference decrease as the number of observations increases, so that

can be regarded as an approximate value of $$\mu.$$ Consequently, the value of $$\mu$$ provided by the erroneous method we discussed in the previous article, must be increased by the ratio of $$\sqrt{\pi - \rho}$$ to $$\sqrt{\pi}.$$

39.
To clearly understand the extent to which it is permissible to consider the value of $${M}$$ provided by the observations as equal to the exact value, we must seek the mean error to be feared when $$\tfrac = \mu^2 .$$ This mean error is the square root of the average value of the quantity

which we will write as:

and since the average value of the second term is evidently zero, the question reduces to finding the average value of the function

If we denote this average value by $${N},$$ then the mean error we seek will be $$\sqrt{\frac{{(\pi - \rho)}^2} - \mu^4} .$$

Expanding the function $$\Psi,$$ we see that it is a homogeneous function of the errors $$e,$$ $$e',$$ $e,$ etc., or equivalently, of the quantities $$\varepsilon,$$ $$\varepsilon',$$ $\varepsilon,$ etc.; therefore, we will find the average value by:

1. Replacing the fourth powers $$e^4,$$ $${e'}^4,$$ ${e''}^4,$ etc., by their average values;

2. Replacing the products $$e^2{e'}^2,$$ ${e}^2{e}^2,$ etc., by their average values, that is, by $$m^2{m'}^2,$$ ${m'}^2{m}^2,$ etc.;

3. Neglecting products such as $$e^3{e'},$$ $e^2{e'}{e''},$ etc.. We will assume (see art. 16) that the average values of $$e^4,$$ $${e'}^4,$$ ${e}^4,$ etc., are proportional to $$m^4,$$ $${m'}^4,$$ ${m}^4,$ etc., so that the ratios of one to another are $$\tfrac{\nu^4}{\mu^4},$$ where $$\nu^4$$ denotes the average value of the fourth powers of the errors for observations whose weight is $$=1$$. Thus the previous rules could also be expressed as follows: Replace each fourth power $$\varepsilon^4,$$ $${\varepsilon'}^4,$$ ${\varepsilon}^4,$ etc., by $$\nu^4;$$ each product $$\varepsilon^2{\varepsilon'}^2,$$ ${\varepsilon}^2{\varepsilon}^2,$ etc., by $$\mu^4,$$ and neglect all terms such as $$\varepsilon^3{\varepsilon'}$$ or $$\varepsilon^2{\varepsilon'}{\varepsilon},$$ $$\varepsilon\varepsilon'\varepsilon\varepsilon'''.$$

These principles being understood, it is easy to see that:

I. The average value of $${\Omega^0}^2$$ is

II. The average value of the product $$\varepsilon^2\,x^0\,\xi^0$$ is

because

Similarly, the average value of $${\varepsilon'}^2\,x^0\,\xi^0$$ is

the average value of $${\varepsilon''}^2\,x^0\,\xi^0$$ is

and so on. Thus the average value of the product

will be

The products $$\Omega^0\,y^0\,\eta^0$$ or $\Omega^0\,z^0\,\zeta^0,$ etc., will have the same average value. Thus the product

will have an average value of

III. To shorten the following developments, we will adopt the following notation. We give the character $$\Sigma$$ a more extended meaning than we have done so far, by making it designate the sum of similar but not identical terms arising from all permutations of the observations. According to this notation, we will have

Calculating the average value of $${x^0}^2\,{\xi^0}^2$$ term by ter, we first have, for the average value of the product $$\alpha^2\,\varepsilon^2\,{\xi^0}^2,$$

Similarly, the average value of the product $${\alpha'}^2\,{\varepsilon'}^2\,{\xi^0}^2$$ is

and so on. Therefore, the average value of the product

is

Now the average value of $$\alpha\alpha'\varepsilon\varepsilon'{\xi^0}^2$$ is

The average value of $$\alpha\alpha\varepsilon\varepsilon{\xi^0}^2$$ is

and so on. Hence, we easily conclude that the average value of the product

is

Thus, for the average value of the product $${x^0}^2\,{\xi^0}^2,$$ we have

IV. Similarly, for the average value of the product $$x^0 y^0 \xi^0 \eta^0,$$ we find

Now, we have

so this average value will be

V. By a similar calculation, we find that the average value of $$x^0z^0\xi^0\zeta^0$$ is

and so on. Adding up, we obtain the average value of the product

this value is

VI. Similarly, we find that

is the average value of the product

and

is the average value of the product

and so on.

Hence by addition we find the average value of the square

which is

VII. Finally, from all these preliminaries, we conclude that

Therefore, the mean error to be feared when

will be

40.
The quantity

which occurs in the expression above, generally cannot be reduced to a simpler form. However, we can assign two limits between which its value must necessarily lie. First, It is easily deduced from the previous relations that

from which we conclude that

is a positive quantity smaller than unity, or at least not larger. The same will be true for the quantity

which is equal to the sum

Similarly,

will be smaller than unity; and so on. Therefore,

must be smaller than $$\pi.$$ Second, we have

since

from which it is easily deduced that

is greater, or at least not smaller, than $$\frac{\rho^2}{\pi}.$$ Therefore, the term

must necessarily lie between the limits

or, between the broader limits

Thus, the square of the mean error to be feared for the value

lies between the limits

so that a degree of precision as great as desired can be achieved, provided the number of observations is sufficiently large.

It is very remarkable that in hypothesis III of art. 9, on which we had formerly relied to establish the theory of least squares, the second term of the square of the average error completely disappears (since $$\nu^4 - 3\,\mu^4 = 0$$); and because, to find the approximate value $$\mu$$ of the average error of the observations, it is always necessary to treat the sum

as if it were equal to the sum of the squares of $$\pi - \rho$$ random errors, it follows that, in this hypothesis, the precision of this determination becomes equal to that which we found, in art. 15, for the determination from $$\pi - \rho$$ true errors.


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 d. upon the gallon of spirits. By increasing the duties upon malt, and reducing those upon the distillery, both the opportunities and the temptation to smuggle would be diminished, which might occasion a still further augmentation of revenue.

It has for some time past been the policy of Great Britain to discourage the consumption of spirituous liquors, on account of their supposed tendency to ruin the health and to corrupt the morals of the common people. According to this policy, the abatement of the taxes upon the distillery ought not to be so great as to reduce, in any respect, the price of those liquors. Spirituous liquors might remain as dear as ever; while at the same time the wholesome and invigorating liquors of beer and ale might be considerably reduced in their price. The people might thus be in part relieved from one of the burdens of which they at present complain the most; while at the same time the revenue might be considerably augmented.

The objections of Dr. Davenant to this alteration in the present system of excise duties seem to be without foundation. Those objections are, that the tax, instead of dividing itself as at present pretty equally upon the profit of the maltster, upon that of the brewer, and upon that of the retailer, would, so far as it affected profit, fall altogether upon that of the maltster; that the maltster could not so easily get back the amount of the tax in the advanced price of his malt, as the brewer and retailer in the advanced price of their liquor; and that so heavy a tax upon malt might reduce the rent and profit of barley land.

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 Journal. Straits Branch, Royal Asiatic Society. No. 62.

Plate V.


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 Plate VI.


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With Plates V. and VI.

The three kĕris illustrated in Plate V, different views of two of which may be seen in Plate VI, are probably the earliest authenticated specimens in any English collection. To describe them briefly. The kĕris on the left, a curving blade with 13 lok, has a hilt showing the features of a European in a helmet and chain gorget, features as caricatured as is the portrait of the Dutchman in the folk-tale of Anggan che Tunggul

Gĕrham-nya ĕmpat sa-rumpun; Sa-gantang makan daging; Dua chupak lĕkat di gigi-nya.

The small kĕris in the centre of Plate V and on the left of Plate VI has a hilt carved with the Javanese representation of some demigod from the wayang kulit cycle, possibly Arjuna; probably, any Javanese could identify it, though writing in England I have no means to discover. It is to be noted that the angle of the hilt follows the conventional attitude of the Jawa dĕmam or cobra-headed Malay hilt, the evolution of which is and probably will remain unsolved. It may be remarked, that the high forehead of Javanese demi-gods would seem to point to this artistic type deriving from a people among whom binding of the head was a custom.

The third kĕris, which is illustrated only in Plate V is interesting for several reasons:—the blade, by the bye, is damasked and has eleven lok. The hilt preserves the conventional angle of the Jawa dĕmam; the long-nosed figure (the nose unfortunately has been broken off) with its row of tooth has a striking affinity with the still longer nosed figure on Patani kĕris hilts, but the back of its head is carved with hair like that of the last kĕris. And it is to be noted that the curved conventional ornament round mouth and chin is almost identical with that round mouth and chin of the Javanese demi-god.

These three kĕris are now in the Ashmolean Museum, Oxford, and belong to the old Tradescant Collection, which dates from the beginning of the 17th century.

John Tradescant the elder is said by Anthony à Wood to have been a Dutchman, but was probably English oven in his name, of which other variants wore Tradeskin and Tredeskin. He and his son of the same name wore travellers, naturalists and gardeners and introduced into England the lilac, acacia, occidental plane and possibly the pine-apple. In the service of George Villiers, Duke of Buckingham, the father dealt with all merchants from all places but especially from Virginia, Bermudas,


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 No tax can ever reduce, for any considerable time, the rate of profit in any particular trade, which must always keep its level with other trades in the neighborhood. The present duties upon malt, beer and ale do not affect the profits of the dealers in those commodities, who all get back the tax with an additional profit, in the enhanced price of their goods. A tax indeed may render the goods upon which it is imposed so dear as to diminish the consumption of them. But the consumption of malt is in malt liquors; and a tax of eighteen shillings upon the quarter of malt could not well render those liquors dearer than the different taxes, amounting to twenty-four or twenty-five shillings, do at present. Those liquors, on the contrary, would probably become cheaper, and the consumption of them would be more likely to increase than to diminish.

It is not very easy to understand why it should be more difficult for the maltster to get back eighteen shillings in the advanced price of his malt, than it is at present for the brewer to get back twenty-four or twenty-five, sometimes thirty shillings, in that of his liquor. The maltster, indeed, instead of a tax of six shillings, would be obliged to advance one of eighteen shillings upon every quarter of malt. But the brewer is at present obliged to advance a tax of twenty-four or twenty-five, sometimes thirty shillings upon every quarter of malt which he brews. It could not be more inconvenient for the maltster to advance a lighter tax, than it is at present for the brewer to advance a heavier one. The maltster does not always keep in his granaries a stock of malt which it will require a longer time to dispose of, than the stock of beer and ale which the brewer frequently keeps in his cellars. The former, therefore, may frequently get the returns of his money as soon as


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 the latter. But whatever inconvenience might arise to the maltster from being obliged to advance a heavier tax, it could easily be remedied by granting him a few months longer credit than is at present commonly given to the brewer.

Nothing could reduce the rent and profit of barley land which did not reduce the demand for barley. But a change of system, which reduced the duties upon a quarter of malt brewed into beer and ale from twenty-four and twenty-five shillings to eighteen shillings, would be more likely to increase than diminish that demand. The rent and profit of barley land, besides, must always be nearly equal to those of other equally fertile and equally well cultivated land. If they were less, some part of the barley land would soon be turned to some other purpose; and if they were greater, more land would soon be turned to the raising of barley. When the ordinary price of any particular produce of land is at what may be called a monopoly price, a tax upon it necessarily reduces the rent and profit of the land which grows it. A tax upon the produce of those precious vineyards, of which the wine falls so much short of the effectual demand, that its price is always above the natural proportion to that of the produce of others equally fertile and equally well cultivated land, would necessarily reduce the rent and profit of those vineyards. The price of the wines being already the highest that could be got for the quantity commonly sent to market, it could not be raised higher without diminishing that quantity; and the quantity could not be diminished without still greater loss, because the lands could not be turned to any other equally valuable produce. The whole weight of the tax, therefore, would fall upon the rent and profit; properly upon the rent of


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, Guinea, Binney, the Amazon and the East Indies for all manner of rare boasts fowls and birds shells and stones." On Buckingham's death, he seems to have entered the service of the king and queen and probably at that date established his physic garden and museum at the oast side of South Lambeth Road, leading from Vauxhall to Stockwell. He was the first in England who "made any considerable collection of the subjects of natural history, He died 1637-1638. The son succeeded the father as gardener to Queen Henrietta Maria, and was also a traveller. In 1656 resolving to take a catalogue of those rarities and curiosities which my father had sedulously collected" he published his Museum Tradescantianum. This book in its 179 pages contains lists of birds, shells, fishes, insects, minerals, fruits, war instruments, habits, utensils, coins and medals. Among entries of other warlike. instruments from Japan. Turkey, India, China, Canada, Virginia, Ginny, Persia, and Muscovy occurs the entry on page 46 of "Poisoned creeses or Daggers, two waved two plain"—apparently one of the plain has been lost. In 1659 Tradescant and his wife sealed and delivered a deed of gift of this collection of rarities to Sir Thomas Ashmole, but his will of date 1661 bequeathed the collection to his wife for life and after her decease to "the Universities of Oxford and Cambridge to which of them she shall think fit." There was a law-suit and Ashmole got the collection "to have and enjoy." In 1677 Ashmole offered it to Oxford as soon as a building should be erected to receive it. In 1683 Wren finished a building and the collection went to Oxford in 'barges', and "the name of Tradescant was unjustly sunk in that of Ashmole." Ashwole, of course, was not a traveller, but Windsor herald and author of a book on the order of the Garter.

The old M. S catalogue of the Tradescant collection compiled in 1685 by Edward Lhywd, first Assistant Keeper of the Ashmolean and still preserved in the archives of the museums contains entries of several creeses vaguely described and an attempt at exact description of the little demi-god hilted kĕris: "117 Pugio anceps in mucronem exit, manubrium simiae imaginem exprimit: a double-edged, dagger tapering to a point,—its handle exhibiting the figure of an ape." The vagina lignea or wooden scabbard has been lost.

The official catalogue of the Ashmolean Collection of 1836 has the entries

32 ''Two Malay creeses or daggers with waved blades. Mus. Tradesc.''

33 ''Another with straight blade. Mus. Tradesc.''

So we have the history of those three kĕris established practically beyond doubt back to the early years of the XVIth century. I am indebted to Mr. E. T. Leeds, formerly of the F. M. S. Civil Service and now Assistant Keeper of the Ashmolean Museum for calllingcalling [sic] my attention to these weapons, getting them photographed for me and putting the Catalogues cited at my disposal,


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 the vineyard. When it has been proposed to lay any new tax upon sugar, our sugar planters have frequently complained that the whole weight of such taxes fell, not upon the consumer, but upon the producer; they never having been able to raise the price of their sugar after the tax higher than it was before. The price had, it seems, before the tax been a monopoly price; and the argument adduced to show that sugar was an improper subject of taxation, demonstrated, perhaps, that it was a proper one; the gains of monopolists, whenever they can be come at, being certainly of all subjects the most proper. But the ordinary price of barley has never been a monopoly price; and the rent and profit of barley land have never been above their natural proportion to those of other equally fertile and equally well cultivated land. The different taxes which have been imposed upon malt, beer and ale have never lowered the price of barley; have never reduced the rent and profit of barley land. The price of malt to the brewer has constantly risen in proportion to the taxes imposed upon it; and those taxes, together with the different duties upon beer and ale, have constantly either raised the price, or, what comes to the same thing, reduced the quality of those commodities to the consumer. The final payment of those taxes has fallen constantly upon the consumer, and not upon the producer.

The only people likely to suffer by the change of system here proposed are those who brew for their own private use. But the exemption, which this superior rank of people at present enjoy, from very heavy taxes which are paid by the poor laborer and artificer, is surely most unjust and unequal, and ought to be taken away, even though this change was never to take place. It has probably been


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 the interest of this superior order of people, however, which has hitherto prevented a change of system that could not well fail both to increase the revenue and to relieve the people.

Besides such duties as those of customs and excise above mentioned, there are several others which affect the price of goods more unequally and more indirectly. Of this kind are the duties which in French are called Péages, which in old Saxon times were called the Duties of Passage, and which seem to have been originally established for the same purpose as our turnpike tolls, or the tolls upon our canals and navigable rivers, for the maintenance of the road or of the navigation. Those duties, when applied to such purposes, are most properly imposed according to the bulk or weight of the goods. As they were originally local and provincial duties, applicable to local and provincial purposes, the administration of them was in most cases intrusted to the particular town, parish or lordship, in which they were levied, such communities being in some way or other supposed to be accountable for the application. The sovereign, who is altogether unaccountable, has in many countries assumed to himself the administration of those duties and though he has in most cases enhanced very much the duty, he has in many entirely neglected the application. If the turnpike tolls of Great Britain should ever become one of the resources of government, we may learn, by the example of many other nations, what would probably be the consequence. Such tolls, no doubt, are finally paid by the consumer, but the consumer is not taxed in proportion to his expense, when he pays, not according to the value, but according to the bulk or weight, of what


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Page:Journal of the Straits Branch of the Royal Asiatic Society (IA journalof606319111912roya).pdf/417

 Population of the Straits Settlements and Malay Peninsula during the last Century

By H. MARRIOTT.

I have collected in the annexed tables statistics of the popula- tion according to race and sex for the Colony, and according to sex for the Malay States.

So far as the Colony is concerned the figures are probably ac- curate from 1871 onwards. Before that date the censuses were taken by the police at somewhat irregular intervals and are not very reliable.

Between 1822 and 1836 the figures have been derived from Mr. Newbold's "British Settlements in the Straits of Malacca." From 1836 to 1860 I have been unable to get many figures, and from that date they are taken from the Census Reports. With regard to the Federated Malay States, no reliable data are to be procured previous to 1891, and for the other Protected States the Census Returns for 1911 are the first of any practical importance.

It may however be of interest to add here the following esti- mate of the population of the States of the Peninsula in the years 1835-1836 given by Mr. Newbold in the work quoted above.


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 he consumes. When such duties are imposed, not according to the bulk or weight, but according to the supposed value of the goods, they become properly a sort of inland customs or excises, which obstruct very much the most important of all branches of commerce, the interior commerce of the country.

In some small States duties similar to those passage duties are imposed upon goods carried across the territory, either by land or by water, from one foreign country to another. These are in some countries called transit duties. Some of the little Italian States, which are situated upon the Po, and the rivers which run into it, derive some revenue from duties of this kind, which are paid altogether by foreigners, and which, perhaps, are the only duties that one State can impose upon the subjects of another, without obstructing in any respect the industry or commerce of its own. The most important transit duty in the world is that levied by the king of Denmark upon all merchant ships which pass through the Sound.

Such taxes upon luxuries as the greater part of the duties of customs and excise, though they all fall indifferently upon every different species of revenue, and are paid finally, or without any retribution, by whoever consumes the commodities upon which they are imposed, yet they do not always fall equally or proportionally upon the revenue of every individual. As every man's humor regulates the degree of his consumption, every man contributes rather according to his humor than in proportion to his revenue; the profuse contribute more, the parsimonious less, than their proper proportion. During the minority of a man of great fortune, he contributes commonly very little, by his consumption, toward the support of that


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Page:Freemasons' circular regarding the laying of Huddersfield Station foundation stone - 1846.jpg

Lodge of Harmony, No. 342,

Masonic Hall, South Parade,

Huddersfield, September 25th, 1846.

,

I am instructed to inform you that the Freemasons have been honoured by the invitation of the to assist in the above ceremony, and that, the proper Masonic Authorities having been consulted, it has been determined to accept that invitation.

having also been specially invited by the Directors, as Co-Trustee with, to the Ramsden Estate; and an invitation of the three Lodges in Huddersfield having been also forwarded, through the D.P.G.M. of West Yorkshire, to the former Nobleman, as Most Worshipful Grand Master of England, it is earnestly hoped that his Lordship may be induced to attend in that capacity. The Circulars of the Grand Lodge, or Provincial Grand Lodge, as the case may be, will be necessarily delayed until his Lordship’s determination be known.

I may however state, for your government, that the Provincial Grand Lodge of Benevolence will be opened for the transaction of Business on the occasion. Also, that immediately after the ceremony of laying the Stone, there will be a Public Cold Collation in the Philosophical Hall. , Junior, Esq. M.P. in the Chair. Tickets, including Wine, 7s. 6d. each. As the number of Tickets must be limited to the capabilities of the Hall, and as the event is regarded generally with great interest, to such Brethren as may wish to be present an immediate application for Tickets is suggested, either to Mr. J., Bookseller, Huddersfield, or to myself.

I am specially desired to request the attendance of yourself, and as many of the Officers and Members of your Lodge as can make it convenient; and that you will be kind enough to make this communication known to them without delay.

I am, Sir and Brother,

Yours fraternally,

THOMAS HEATON,

Secretary.


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 State from whose protection he derives a great revenue. Those who live in another country contribute nothing, by their consumption, toward the support of the government of that country, in which is situated the source of their revenue. If in this latter country there should be no land tax, nor any considerable duty upon the transference either of movable or immovable property, as is the case in Ireland, such absentees may derive a great revenue from the protection of a government to the support of which they do not contribute a single shilling. This inequality is likely to be greatest in a country of which the government is in some respects subordinate and dependent upon that of some other. The people who possess the most extensive property in the dependent, will in this case generally choose to live in the governing country. Ireland is precisely in this situation, and we cannot therefore wonder that the proposal of a tax upon absentees should be so very popular in that country. It might, perhaps, be a little difficult to ascertain either what sort, or what degree of absence would subject a man to be taxed as an absentee, or at what precise time the tax should either begin or end. If you except, however, this very peculiar situation, any inequality in the contribution of individuals, which can arise from such taxes, is much more than compensated by the very circumstance which occasions that inequality; the circumstance that every man's contribution is altogether voluntary; it being altogether in his power either to consume or not to consume the commodity taxed. Where such taxes, therefore, are properly assessed and upon proper commodities, they are paid with less grumbling than any other. When they are advanced by the merchant or manufacturer, the consumer, who finally pays them,