text:
Translation:Ayil Meshulash/Discourse 5

This chapter contains sections 140 to 162

Section 140
If one marks a point within a circle outside of center, like so:

where point A is the center and point B is the point outside the center, the greatest line that can be drawn from this point towards the circumference of the circle (point C) will pass through the center. This line is called the maximum [WHAT IS THE CORRECT TERM?], for its end at the circumference is the farthest point in the circle from point B. If the rest of this line is completed as a diameter across the circle the point on the other end (point E) is called the minimum [WHAT IS THE CORRECT TERM?], for it is the closest point in the circle to point B, and a line drawn from point B to point E is the shortest line that can be drawn from that point until the circumference. The proof is that any line drawn from point B until some other point, like point D, for example, must be shorter than line BC, as lines AD and AB together are longer than line BD, and line AD is equal to line AC. If so, line AD and line AB together are equal to line BC. Since line BD must be shorter than line BC, and line BD is longer than BE, since line BD and AB together are more than line AD, as we have shown, and Line AB and BE together are equal to line AD, line BD therefore must be greater than line BE, and all other lines extending from point B that travel away from the direction towards the upper or lower bounds are by force becoming smaller.

Section 141
If line AB is divided into two equal parts, and a right angle line is extended until the circumference (at point F), this line is greater than half the diameter, all lines extending from point B to a point between the maximum and point F are greater than half the diameter, while all lines drawn from point F to the minimum (under point B) is less than half the diameter. A line drawn from point B to point F is half the diameter. Since line FG divides line AB in two parts at the right angle, the triangle ABF is an equal leg triangle (isosceles), as was written in Section 52, line AF is half the diameter, and therefore line BF is half the diameter as well.

Section 142
Angle CAD is greater in all possible lines extending from the diameter [from point A] than angle CBD is in all the possible lines extending from point B when they both meet at the same point on the circumference, as was written in section 67. Similarly, angle DAE is smaller than angle DBE in all its possible paths, as was written there.

Section 143
The difference [in degrees] between angle CAD and CBD is equal to angle ADB. The proof to this is to draw a line parallel line to AD from point B until the circumference, like so:

Here line BF is parallel to line AD, and angle CBF is equal to angle CAD, as was written in section 20. If the angle CBD is subtracted from these angles the remainder is angle DBF, and DBF is equal to angle ADB, as was shown in section 62.

Section 144
This can be proven through an alternate method as well. As angle CAD is equal to the sum of angles ABD and ADB, as was written in section 65. This being so, angle CAD is greater than angle ABD in the amount of angle ADB, and the same result will always result when lines are drawn from along the diameter line from the maximum to the minimum, and this will be the difference between angles DAE and DBE in every arrangement [as angle DBE contains the sum of the degrees of angles DAB and ADB].

Section 145
If the distance between the center and the point along the diameter is known (line AB in above figure), it is possible to determine the angle in degrees of angle ADB for every degree from the maximum to the minimum [top of diameter to the bottom]. In addition, it is possible to determine length of line BD wherever it may be, provided it extends from the point to the surface of the circle. This is due to the fact that a) Angle CAD is known, b) Angle DAB is known, as it is the remainder [to 180 degrees], as was written in section 15, c) Line AD half the diameter, and d) Line AB is known. With these values known the other quantities can be determined, as was written in section 133. Similarly, if the opposite angle [ADB] is known, the distance between the center and the point can be determined.

Section 146
The opposite angle [AIB] grows larger as it sweeps from the maximum to the minimum. At point I, the line extending to point B on diameter line CD forms a right triangle, and as the angle sweeps past point I it becomes increasingly smaller. Therefore, point I is point where the sweeping angle's measure is the greatest.

The proof of this is that the proportion of line AI to line AB is equal to the proportion of the sine of angle ABI to the sine of angle AIB [text corrected per commentary]. This, and the fact that lines AI and AB are always the same length during the sweep cause the sine of angle AIB to enlarge if the sine of angle ABI is enlarged. Since the measurement of the sine of a right angle is the greatest of all angles, as was written in section 83, and this is angle ABI at point I, angle AIB and point I is the greatest of all opposite angles along the sweep.

Additionally, the measure of the sine of angle AIB at point I [which is line AB, to the sine of angle ABI, line BI] is equivalent to the proportion of line AB to half the diameter, as was written in section 87. [IS THIS CORRECT?]

The situation is reversed when sweeping from the minimum to the maximum, where from point D to I the opposite angle increases and from point I to the maximum it decreases.

Section 147
If one extends lines from the center of the circle to every possible angle along the circumference of the circle, and compares them to lines extended from a point removed from the center to every degree [and compares the two sets of lines,] one will find that from the maximum to point I [in previous section] the angle created by the center point always exceeds the angle created by the other point, while from point I to the minimum the angle created by the other point always exceeds the angle creates by the center point. For example, angle EAF is greater than angle EBF, and angle GAH is less than angle GBH.

Section 148
The closer an arc is to the maximum the greater will be the difference between two angles, such as EAF and EBF [created by angle formed from the each end of the arc to the center and a point outside the center]. As the arc sweeps downward the difference between the two angles decreases until point I. Similarly, the closer an arc is the minimum the greater the difference between two angles, for example GAH and GBH, and that diminishes as the arc continues to sweep to point I.

Section 149
Everything we have mentioned regarding the area to the left of the diameter CAD is true for the right of the diameter as well. Therefore, if you calculate 180 degrees of angle and arc from the maximum you will reach the minimum, and from 180 degrees to 360 degrees you will pass from the minimum to the maximum.

Section 150
If you draw a small circle that passes through the circumference of the large circle and then draw a line from the center of the large circle across the small circle to its circumference, the longest line that can be drawn is on that passes through the small circle’s center, like so:

Here, where circle ABCD is drawn over the circumference of circle ADK, the longest line that can extend from the center at point H until the circumference of the small circle is line HB. Point B is called the maximum. The shortest line that can be drawn to the small circle from the center is line GH, and point G is called the minimum. The proof to this rests on what was mentioned in section 140, that if a line such as HC is drawn it is perforce smaller than line HB, since lines HF and HE are equal, and lines BF and CF are equal, and the sum of lines HF and CF is greater than line HC. Since this is so, line HB is also greater than line HC, and line HC is greater than line HG. The proof to this is similarly found by referring to that section, for the sum of lines HC and FC is greater than line HF and greater than the sum of lines HG and CF, which is equal to line HF. This follows the idea that all lines drawn out [as one sweeps around the circumference] from the maximum to the minimum decrease in length. All the rules explained from section 140 to section 149 can be found in this example, as will be explained.

Section 151
[In the figure above,] all the lines that extend from the maximum until the circumference of the large circle between points D and A are all larger than half of the diameter of the large circle, which is line HF in the figure above. From between these points [along the circumference] until the minimum all lines will drawn to the minimum be shorter than half of the diameter, and at these two points lines drawn would be equal to half the diameter, as was written in section 21. [THE COMMENTARY IS UNABLE TO RELATE THIS SECTION TO SECTION 21, AND SUGGESTS THAT SECTION 7 MAY BE THE CORRECT TEXT.]

Section 152
In the figure above, the difference between angle BFC and angle CHF is angle HCF. The proof to this is also follows from what was mentioned in section 143, and the same is true for every point along the circumference of the small circle, where the angle continually grows [as the point moves] from the maximum to the minimum. Angle HFC then has the opposite property, for it decreases as it sweeps from the maximum toward the minimum. At the same time, angle CHF increases as it sweeps from the maximum to the point where a line extending from the center of the small circle forms a right angle with a line that extends from the center of the large circle. This occurs at point I. The proof of this also follows section 146, for the proportion of half the diameter of the large circle to half the diameter of the small circle is equivalent to the proportion of the sine of angle FCH to the sine of angle CHF. Therefore, since the halves of the diameters of each circle always measure the same lengths respectively, the size of angle CHF is wholly dependent on the change of the sine of angle FCH. Now the greatest of all sines is that of the right angle [which angle HCF forms], and if so, the angle CHF must be the largest of all the angles. This means that from that point on they decrease until the minimum is reached, while from the minimum to the maximum the opposite is true.

Section 153
If one knows the proportion of the diameter of the small circle in relation to the diameter of the large circle, and also knows how many degrees away a point is [along the circumference] from the maximum, which [in our example] is angle BFC, one is also able to determine all three angles of the triangle CHF, and the length of side FH, since angle CFH is known to be the remainder of angle BFC. In addition, the lengths of sides CF and HF are then known. Since this is so, the other quantities can be determined as well. Similarly, if one knows the proportion of the diameters and angle CHF [and angle BFC is unknown], the remaining quantities can be determined. Furthermore, if angle BFC and CHF are known and the diameter of the large circle is also known the other quantities can be determined, and the same is true if the other quantities are known.

Section 154
It is also clear from what was explained in section 152 that regarding angle CHF, when it is at its greatest point [and angle HCF is a right angle, so that line FC reaches point I,] the proportion of half of the diameter of the large circle [line HF] to half of the diameter of the small circle [line CF] is equivalent to the proportion of the sine of angle HCF to the sine of angle CHF.

All the principles explained [to this point] regarding a point lying distant from the center relate to the figure outlined in section 150.

Section 155
We have previously explained that through every unit of change from the maximum until point I the angle CHF increases. However, the increase in degrees per unit change is not constant. The closer to the maximum the greater the angle measure. The same is true for the minimum. In this example,

the arc points are B, K, and C. When drawn further out as another arc with the center at H and the radius extending to B, arcs BK and KC are equal in arc measure and lines drawn out from point H would create an equal pair of angles. However, these same changes of degree on the arc of the small circle would create pair of angles that are do not equal each other when the lines are drawn out from point H. When lines are drawn out to these points on the small circle from point H, angle BHK is greater than angle KHC, [and the rate of change in the increase of angle CHF is diminishing as it sweeps left]. The same is true for the minimum, that though arcs GL and LM are equal, angle GHL is greater than angle LHK. All that has been written in regard to the small circle is true for the right side of this circle as well. Therefore, if 180 degrees are calculated from the maximum the above can all be calculated from the maximum to the minimum, and if 180 degrees are added from that point until 360 degrees the same calculations can be made from the minimum to the maximum. '''[THE TRANSLATION OF THIS SECTION IS UNCLEAR. IT HAS BEEN RENDERED IN A WAY THAT SEEMS SENSIBLE, BUT IT MAY OR MAY NOT BE AN ACCURATE TRANSLATION OR CORRECT.]'''

Section 156
If the [circumference of the] circle lay outside the center of the circle, similar to [the idea depicted in] section 140 [UNSURE OF THIS PHRASE],  and it is equal to the large circle in the previous section, and the difference between the center and a point outside the center is equal to half the diameter of the small circle: Since this circle is equal to the circle in the previous section and line TS here is equal to half the diameter, which is line BF, all the six quantities of triangle TRS are equal to the six quantities of triangle CFH in every degree as R becomes distanced from N, and C becomes distanced from B, respectively, if the degree changes are equal. Angles CFH and RTS are equal, for they are the remainder of arcs BC and NR. Lines FH and TR are also equal [both are radius lines], and lines CF and TS are equal as well. Therefore, all the other quantities [of these two triangles] are equal, as was written in section 24.

Section 157
If angle VHM is equal to angle EMP and line HY is equal to half the diameter of the small circle which is line MP, then it follows that line YP is equal to line HM. The proof of this is that since angles VHM and EMP are equal the result must be that lines YH and PM are parallel, as was written in section 20. Consequently, angles PHY and HPM are equal, as was written in section 62, and lines HY and PM are equal, with line PH being shared by the two triangles PYH and PMH. If this is so, all the quantities of these triangles are equal, as was written in section 24.

It further follows that lines YP and HM are parallel since angles YPH and PHM are equal, as was written in section 62. In addition, it is evident that the angles VYP and VHM are equal, since the lines YP and HM are parallel. All of this is true in regards to all the parts of the large circle in which is set the center of a small circle, for point M in relation to point V in the large circle is as the relation of point P to point E in the small circle, and as a result the two arcs VM and PE are equal.

Section 158
From here it is apparent that if a small circle is inscribed so as to overlap a large circle, and point P is placed on the circumference of the small circle along the same vertical line as the center of the small circle, and then this point P is used to draw a another circle, such that the new circle is the same size as circle ABKCZ and its center is at point Y (as it has been explained in the previous section that line YP is equal to line HM, which is half the diameter), then the following is so: Point Y is the center, point H is outside the center, the triangle YPH in the new circle is completely identical to triangle TRS [in section 156] in all its particulars within circle NRE [Where is the ‘Hey’ in the diagram?]. This is because lines YH and TS are equal, lines YP and TR are equal, and angles PYH and RTS are equal. These angles are also the respective remainders of angles VYP and NTR. Therefore, all the respective quantities of these triangles are equivalent. The same is true for every part of these circles, so that the new circle can be placed on circle NRE [Where is the ‘Hey’ in the diagram?] in an orientation that would allow triangle YPH to be exactly positioned over triangle TRS. The same maximum and minimum points [along the diameter] that connect the lines for TRS would also connect the lines for YPH and are therefore equal, with lines PY and PH corresponding to lines RT and RS, providing that the degrees measure from point M to point V on the left, and point R from point N on left, and point P from point E on the right are equal, or alternatively, the degree measure between point M and points R (on the right) and P (on the left) are the same. [NOT SURE WHAT THE REFERENCE TO ‘RIGHT’ AND ‘LEFT’ MEAN]

Section 159
All of the above applies to lines extending from the center. If, however, the lines extend from a point outside the center, arcs that are of equal measure in two different locations on a small circle will not create equal angles. In this figure,

lines TY, BY, HY and VY all extend from point Y, which is removed from center P. While arcs AB and HV are equal, angle HYV is nevertheless greater than angle AYB. Similarly, as the distance from the maximum increases so does the angle measure, up until minimum point M where the angle is at its largest measure. From the minimum until the maximum the angle decreases until it reaches the maximum, where it is at its smallest measure. [NOT SURE OF THIS SECTION]

Section 160
To determine this initial angle measure (of angle HYV), one must know the measure of the line that extends from point Y until the center of the small circle, which is line CY. This is known by use of the triangle PYC, as was explained in section 145. When line HC and angle HYC are known the remaining quantities of triangle HCY [per commentary] can be determined.

[Commentary: This means to say that after one has determined angle APC, which presupposes that the angle measure between center C of the small circle and point T of the large circle is known, then one can determine angle CPY, which is the unknown angle’s (APC’s) remainder. Now line PC is known to be half the diameter, and line PY can also be known through the application of section 145. If all this is so, line CY can be determined and line HC is also known, as it is half of the diameter of the small circle. Angle HCY can also be known, as we have seen it is the remainder of angle HCV, for we know the angle measure between point H and V in the small circle. Therefore, the angle HYV [the angle this section is interested in] can be determined.]

Section 161
An alternative method that can be used, providing lines PY and HC are equal: Set the following proportion: The sine of angle CPY compared to the sine of angle PCY is proportional to the sine of angle CHY to the sine of angle CYH. Now angles CPY and PCY can be known, as was written in section 145. Once the proportion of angles CHY and CYH are known, and the summation of angles CHY and CYH are known to be equal to angle HCV (as was written in section 65), then the measure of each of the angles CHY and CYH can be determined (as was shown in section 95) [and angle CYH is equal to angle HYV].

The proof of this is that the sines are proportional to each other, for as is the proportion of line CY to line PY so is the proportion of the sine of angle CPY to the sine of angle PCY (as was shown in section 87). Furthermore, the proportion of line CY to line HC is as the proportion of the sine of angle CHY to the sine of angle CYH. The line PY of this proportion is then equal to the line HC of the other proportion. Therefore, proportion of these pairs of angles must be equal to each other [CPY:PCY = CHY:CYH].

Through this approach any arc on the small circle can be used to determine the degree measure of any arc on the large circle from the maximum to the minimum and from the minimum to the maximum [through 360 degrees].

Section 162
If two lines extend, one from the center of the large circle and one from a point outside the center, and both pass through the center of the small circle until its circumference, like so,

where lines BH and CH extend respectively from center B and point C to center H: In this formation one can determine arc GI at any degree of the large circle providing on knows the measure of arc AH. In addition, the following can be known: line BC, line BH (which is half of the diameter), angle BHC, and that angles BHC and GHI are equal. Know further that all I have written in regard to a point that is outside the center from section 140 to 150 will apply to angle GHI here since angles BHC and GHI are equal and triangle BHC is equal to triangle ADC from section 140. The same is true for angle FHI that comes about through lines BH and DH and for the two lines extending from points outside the center, lines CH and DH, which create angle FHG. However, in this later instance either triangle CHD must be calculated first before triangle BHC is calculated so that line HC can be known, as was written in section 160, or, if lines BC and CD are equal, the proportion of the sines of the angles method may be used, as was written in the previous section.

איל משולש

title: Translation:Ayil Meshulash/Discourse 5


text:
Collier's New Encyclopedia (1921)/White Sea

WHITE SEA, a branch of the Arctic Ocean which extends S. W. into the province of Archangel as far as the 64th parallel. It is separated from the open sea by the peninsula of Kola and a strait 246 miles long, and from 106 miles (between Capes Sviatoi and Kanin) to 28 wide, and divides itself for the most part into three bays, of which the broadest is the Dwina Bay to the S. E., the S. the Onega Bay, and the longest the Kandalak Bay in the W. Its area, including the N. E. bay, into which the

Mezen falls, is 47,346 square miles. The coasts are flat, with numerous lakes, small rivers, and mountains in the N. and E. The White Sea is blocked with ice except during the months of June, July, and August. Canals connect it with the Volga and the Dnieper. The dwellers on its shores are Lapps, Finns, and Samoyedes, who live by fishing, seal hunting, and the chase. The chief port is Archangel. The passage to the White Sea was discovered in 1553 by Richard Chancellor, an officer under Sir Hugh Willoughby. The English in 1584 established the little fort of Archangel as the center of the White Sea trade, in which they enjoyed great privileges till the founding of St. Petersburg.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text: Popular Science Monthly/Volume 67/August 1905/An Eclipse Observer's Experiences in Sumatra


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Rh ILLUSTRATED INSTRUCTIONS. In the pattern envelope you will find THE DELTOR in which are the BUTTERICK ILLUSTRATED INSTRUCTIONS. Look these over and see how easily your dress will go together.

THE ILLUSTRATED INSTRUCTIONS show you with a series of pictures how to join every part of the garment, just where to baste, tuck, drape, etc. You do not have to read long, confusing directions for it is all told in pictures which, with a few explanatory words, are impossible to misunderstand. You see at a glance what you are to do just as if there were someone at your work-table putting your garment together for you.

THE ILLUSTRATED INSTRUCTIONS show you just how to use every perforation and every notch. If you have never used a pattern in your life the ILLUSTRATED INSTRUCTIONS make it possible for you to make any type of garment without any knowledge of dressmaking, because the knowledge is supplied you by an expert who has reduced it to pictures.

OUTLET SEAMS are marked by large single perforations (Ill. 23.) In basting them the basting line should run exactly through the center of these perforations. (Ill. 23.)

Ordinary seams are not marked by perforations but are basted exactly ⅜ inch from the seam edge. (Ill. 24.)

The outlet seam is deeper than the ordinary seams. It is made so on purpose so that it can be let out if it is necessary to make any slight alteration to suit the individual figure. They are generally used at underarm and shoulder seams, and very often in the seams of sleeves. In so many cases women's shoulders are not exactly even or there are some slight variations from the average at one point or another of the figure. These outlet seams give you a chance to alter the garment in an easy, simple way.

ORDINARY SEAMS. A ⅜ of an inch seam allowance is made on all edges not cut on the fold of the goods, or finished with a hem. In basting, the seam lines must be followed exactly. (Ill. 24.) If you make them deeper or narrower you will alter the size of the garment.

DARTS are marked by V-shaped lines of perforations.

A dart is made by bringing the two lines of a dart perforations together and basting through the perforations. (Ill. 25.)

title: Page:Newdressmakerwit00butt.djvu/21


text: Popular Science Monthly/Volume 67/August 1905/Public Interest in Research


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18 Illustration 25 shows the easy curve which should be followed in finishing a dart in a waist pattern. The illustration shows the material basted with the corresponding perforations matched according to pattern instructions. The line of the basting should follow a reverse curve toward the point, running into the fold almost on a line with the fold. If a reverse curve is not followed (III. 26.) you will get a pouch effect, a sort of pucker, something that is seen at the top of a badly sewed dart.

All perforations and notches should be followed faithfully. They are the work of experts who have gone to an immense amount of labor and study to show you the absolutely correct way of putting together your garment. It takes a little time to mark them all carefully in the beginning, but you save that time over and over again before you are finished.

All the working perforations should be marked with tailors' tacks, using different colored cotton to mark the different sizes and kinds of perforations, so that you won't confuse them after you have removed the pattern from the material. (Chapter 16 III. 146.)

THE DELTOR gives complete suggestions for finishing the garment in the different kinds of material to which it is suited. The finishes given in the are those that would be used by the best Paris and Fifth Avenue establishments and if they are followed carefully the garment will have a well-made French look instead of "the home-made look" which is the result of improper and careless finishing. The suggestions given in the are explained and illustrated in. For example, if the tells you to use a French fell seam or a bound buttonhole, you will find illustrated directions for making the seam and the buttonhole in.

NOTE: If after reading this chapter carefully and following the instructions you are not entirely satisfied with the way you ha'e made any style of garment from a Butterick Pattern write to Eleanor Chalmers in care of the Butterick Publishing Company, New York City. Explain your difficulty as fully as you can and Mrs. Chalmers will help you.

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text:
Translation:Ayil Meshulash/Discourse 6

This chapter contains sections 163 to 189

Section 163
The definition of the tangent and the secant: If a line that is half the diameter in length is extended from the center to one end of an arc, and at this point another line is drawn outside of the circle vertically at a right angle to this line, the vertical line is called the 'tangent'. A line drawn from the center of the circle to the other end of the arc, extending in such a way that it eventually meets the vertical line is called the secant. In this example: Line BC is half the diameter, line AC is the vertical tangential to the arc GC, and line AB is the secant of arc GC. There already exists a chart of all tangents and their secants for every degree in all arcs and also angles - since tangents and secants of arcs are also tangents and secants of angles, and follow the same patterns as written for sines.

Section 164
A right angle or quarter arc has no tangent or secant since the secant drawn outward will never meet the vertical. For example, in arc FC above, the tangent CA and the secant BF never meet since they are parallel lines, as was written in section 19.

Section 165
The tangent and secant are 'partners' with an arc and its remainder. For example, tangent AC and secant AB are 'partners' with arc GC and arc GH. They also have the same relationship with the angle and its remainder.

Section 166
The complementary tangent and secant are the tangent and secant that complete the arc to 90 degrees. If the arc is greater than 90 degrees, the degrees of complementariness are the tangent and secant of an arc with a measure of degrees over and above 90. For example, the tangent FI is the tangent for arc FG, which is complementary to the tangent of CA, which is the tangent for arc CG and arc HG. Similarly, secant BI is complementary to secant BA for the same reason.

Section 167
The ratio of the tangent to the maximum sine is proportional to the ratio of the maximum sine to the complementary tangent. The proof to this can be seen in the previous figure, where the ratio of [vertical tangent] line AC to [length of maximum sine, half the diameter] line BC is proportional to the ratio of [maximum sine] line BF to [complementary horizontal tangent] line FI. [Refer to section 92]

Section 168
The ratio of the tangent to the secant is proportional to the ratio of the maximum sine to the complementary secant. For example, the ratio of line AC to line AB is proportional to the ratio of line BF to line BY. Similarly, the ratio of the maximum sine to the secant line is proportional to the ratio of the complementary tangent line to the complementary secant line. For example, the ratio of line BC to line AC is proportional to the ratio of line AF to line BI.

Section 169
The ratio of the complementary sine to the sine is proportional to the ratio of the maximum sine to the tangent. The proof of this is as follows: line BD is equal to line ED, which is the complementary sine of arc GC. Line GD is the sine of arc DB, and lines GD and AB are parallel since they are both right angles from line BC. Since this is so, the ratio of line BD to line GD is proportional to the ratio of line BC to line AC. Furthermore, the ratio of the sine to the complementary sine is proportional to the ratio of the maximum sine to the complementary tangent line, for if one compares BG to arc FG one will find that line BD is the sine of the arc, and line GD is the complementary sine, with line AG being the complementary tangent line.

Section 170
The ratio of the cosine to the maximum possible sine line is equivalent to the ratio of the maximum sine line to the secant that divides the arc The proof of this is similar to what was written in a previous section [92] where the proportion of all these are explained in relation to arc FG. [Referring to the figure in section 163, the arc FG creates sine EG and cosine GD. These together create two triangles facing each other, BFI and BGD. These two triangles were shown to be in proportion to each other, so that vertical line GD is to vertical line BF as diagonal BG is to diagonal BI.]

Section 171
The proportion of the sine to the maximum sine line is equivalent to the proportion of the tangent to the secant. The proof of this is that the proportion of line GD to BG is equivalent to the proportion of line AC to AB.

Section 172
The square of the secant is equal to the sum of the squares of the maximum sine line and the tangent line. In the above figure the square of AC is equal to the sum of the squares of lines BC and AB, as was written in section 75.

Section 173
In right triangle [ABC], the proportion of the tangent [AC] of angle A to the maximum sine [BC] is equivalent to the proportion of the line opposite angle A [line GD] to the line perpendicular to it [line BD]. Similarly, the proportion of the maximum sine [BC] to the secant of this angle [BA] is equivalent to the proportion of the line adjoining the angle [BD] to the hypotenuse [BG]. The proof of this can be found in triangle ABG, where line BC is considered the maximum sine, and an arc drawn with this diameter [with its center at point A] would form arc GC, with angle ABC having the tangent of AC and the secant AB. Therefore, if line AC [per commentary] is considered the maximum sine then side BC would become the tangent of angle BAC and line AB would be its secant.

Section 174
[Returning to the methods of deducing a triangle's quantities...] Now see before you that there is another method whereby you may know all remaining quantities [aside from those previously discussed starting in section 110], if the figure is such that two quantities aside from the right angle are known in a right triangle. This approach will make use of tangents and secants. The first method: If the lengths of the vertical and base are known, meaning lines AB and BC are known, the proportion of side AB to side BC will be found to equivalent to the proportion of the maximum sine to the tangent of angle BAC. Once the angle BAC is known, realize that a) the proportion of the maximum sine to the secant of angle BAC is equivalent to the proportion of side AB to AG, b) that the proportion of the tangent of angle BAC to the secant of BAC is equivalent to the proportion of side BC to AC, and c) that the proportion of side BC to AB is equivalent to the proportion of the maximum sine to the tangent of angle ACB. Following this one may calculate that a) the proportion of the maximum sine to the secant of angle ACB is equivalent to the proportion BC to AC, and b) that the proportion of the tangent of angle ACB to the secant of ACB is equivalent to the proportion of side AB to side AC.

Section 175
The second method: If the length of the vertical line and the hypotenuse are known, for example, if lines AB and AC are known, then the proportion of line AB to line AC is equivalent to the proportion of the maximum sine to the secant of angle BAC. Following this you will find that the proportion of the maximum sine to the tangent of angle BAC is equivalent to the proportion of line AB to line BC, and the proportion of the secant of angle BAC to the tangent of angle BA is equivalent to the proportion of side AC to side BC.

Section 176
The third method: If the base line and the hypotenuse are known, for example lines AB and BC, then base can be set up in proportions in the same manner as was done with the vertical side in the previous section.

Section 177
The fourth method: If and BAC and side AB are known, recognize that the proportion of the maximum sine to the tangent of angle BAC is equivalent to the proportion of side AB to side BC, or that the proportion of the tangent of angle ACB (which is the cosine of angle BAC) to the maximum sine is equivalent to the proportion of side AB to side BC. Additionally, the proportion of the maximum sine to secant of angle BAC is equivalent to the proportion of side AB to side AC, and the proportion of the tangent of the angle complementary to angle BAC [IS THIS CORRECT?] to the secant of the angle that is complementary to angle BAC [IS THIS CORRECT?] is equivalent to the proportion of side AB to side AC.

Section 178
The fifth method: Angle BAC and side BC are known, as was seen in the previous section, only instead of employing the procedure against angle BAC it is employed with its complementary portion, angle ABC. In this case all that was done with the then complementary angle BAC is now done with angle BAC.

Section 179
The sixth method: If angle BAC and side AC are known, the proportion of the secant of angle BAC to the maximum sine is equivalent to the proportion of side AC to side AB, and the proportion of the secant of angle BAC to the tangent of angle BAC is equivalent to the proportion of side AC to side CB. Alternatively, the proportion of the secant of the complementary angle of angle BAC to the tangent of the complementary angle of angle BAC is equivalent to the proportion of side AC to side AB, and the proportion of the tangent of the complementary angle of angle BAC to the maximum sine is equivalent to the proportion of side AC to side BC.

In all these various methods there is no need to calculate the measure of the third angle, as was seen in a previous section [section 118].

Section 180
If triangle under discussion is not a right triangle, whether it is equilateral, isosceles or neither - the method to use has been shown in a previous section [121], where the vertical side is drawn within triangle or outside it. This enables one to calculate the quantities of the right triangles created via use of the tangents and secants.

Section 181
[To explain:] If the right angle line is drawn within the triangle, the proportions of the tangents of the intersecting angles are equivalent to the proportion of the sides opposite them. For example, in this figure:

The proportion of the tangent of angle BAD to the tangent of angle DAC is equivalent to the proportion of side BD to side DC. The proof of this is that the proportion of the maximum sine to side AD is equivalent to the proportion of the tangent of angle BAD to side BD, as was written in section 173. Similarly, the proportion of the maximum sine to side BD is equivalent to the proportion of the tangent of angle DAC to side DC. This being so, it follows that the proportion of the tangents of the angles to these angles' opposite sides are also equivalent, as was written in section [36, per commentary].

Section 182
If the vertical line falls outside the triangle, as in this figure:

Then as is the proportion of the tangent of angle BAC to the tangent of angle CAD, so is the proportion of side BC to side CD. The proof of this is that the proportion of the maximum sine to side AD is equivalent to the proportion of the tangent of angle BAD to side BD. In the same fashion, the [proportion of the maximum sine to side AD is equivalent to] the proportion of the tangent of angle CAD to side CD. Since this is so, the proportion of the tangents of these angles to sides opposite these angles are also equivalent.

Section 183
[Relating to section 181,] To continue, the proportion of the tangents of the angles opposite the vertical side are equivalent to the proportion of the sides that serve as secants. For example, as is the proportion of the tangent of angle ABC [to the tangent of angle ACD] so is the proportion of side DC to side BD. The proof to this is that the proportion of side CD to the maximum sine is equivalent to the proportion of side AD to the tangent of angle ABC, as was written in section 173. Similarly, the proportion of side DC to the maximum sine is equivalent to the proportion of side AD to the tangent of angle ACD. Now realize that in both of these proportions the middle terms are equivalent, and if so, the two extreme terms are proportional to each other as well, as has been shown in section [36].

Section 184
If the vertical falls outside of the triangle, as was depicted in section 182, there is also of the equivalence shared by the proportion of the tangent of angle ABC to the tangent of angle ACB [ACD, according to commentary] and the proportion of side DC to side BD. The proof of this can be seen in the previous section as well.

Section 185
The proportion of the secant of angle BAD to the secant of angle DAC is equivalent to the proportion of side AB to side AC. The proof of this is that the proportion of the maximum sine to side AD is equivalent to the proportion of the secant of angle BAD to side [AB, per commentary]. So too is the relationship of the secant of angle DAC to side AC. Therefore, these two secants themselves share an equivalency.

Section 186
Continuing with the scenario where the vertical falls outside the triangle, there is equivalence between the proportion of the secant of angle BAD to the secant of angle DAC and the proportion of side AB to side AC. The proof to this is also based on the previous section.

section 187
In addition, [based on section 181,] the proportion of the tangent of angle ABC to angle ACB is equivalent to the proportion of the tangent of angle DAC to the tangent of angle BAD, for just as the proportion of side DC to side BD so is the proportion of the tangent of angle ABC to angle ACB, as is written in section [183, per commentary]. The proportion of the tangent of angle DAC to the tangent of angle BAD [is equivalent to the proportion of side DC to side BD, as was written in section 181], if so [once it is seen that DC and BD are equivalent through the proportions the remainders are also equivalent. That means tangent ABC is to tangent ACD or tangent ACB as tangent DAC is to tangent BAD].

When the vertical falls outside the triangle there is another equivalence, and that is the proportion of the tangent of angle ABC to the tangent of angle [ACD, per commentary] and the proportion of the tangent of angle DAC to the tangent of angle BAD. The proof is based on what was written in section [184, per commentary].

Section 188
And now, concerning a triangle that is not a right triangle, one can attain these measurements by different means: If two of the angles and the side between them are known, a right angle may be drawn from the third angle until the side that is known, like so,

where angles ABC, ACB, and side BC are known, and vertical side AD is drawn. Here the proportion of the tangent of angle ABC to the tangent of angle ACB is equivalent to the proportion of side DC to side BD, as was written in section [183, per commentary]. Through this, the three quantities may be known, [as angle ABD may be known, angle ADB is a right angle, and side BD is known as well. The same goes for triangle ADC, and the remaining quantities can also be known, as was written in section 110]. A similar procedure may also be used when the right triangle falls outside the triangle, following what was written in section [184, per commentary].

Section 189
Alternatively, if two sides and the angle between them are known - for example when sides AB, AC, and angle BAC are known, and right angle AD is drawn: The proportion of side AB to side AC is equivalent to the proportion of the secant of angle BAD to the secant of angle DAC [as in section 185]. Through this method, [in triangle ADB angle ADB is a right angle, angle BAD is also known once the proportion of its secant to the secant of DAC is known, and then angle BAC is already known. Since this is so, the two angles that are divided by it are also known. Side AB is known. The same is also true in triangle ADC.]

If the right angle falls outside the triangle, the process would be according to section [186, per commentary].

איל משולש

title: Translation:Ayil Meshulash/Discourse 6


text:
Collier's New Encyclopedia (1921)/Whiting

WHITING, a city of Indiana, in Lake co. It is on the Elgin, Joliet and Eastern, the Indiana Harbor Belt, the New York Central, the Pere Marquette, the Baltimore and Ohio, and the Pennsylvania railroads. Its industries include the manufacture of oilcloth, chemicals, asphalt pavement, etc. It has a public library and a park. It is an important station of the Standard Oil Company. Pop. (1910) 6,587; (1920) 10,145.


text:
Portal:United Nations Security Council Resolutions/2013

联合国安理会2012年决议


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Page:Newdressmakerwit00butt.djvu/167

Rh facing of the cloth to the front of the trousers on the right side. This facing should be the same length and width as the facings on the back edge of the pocket. Stitch the pocket, trousers and facing together in a narrow seam. Turn the facing over on to the pocket and run a row of stitching close to the fold to hold it in place. Turn under the back edge of the facing and stitch it to the pocket. Trim off the lower corners of the pocket (Ill. 368), and crease the edge for a seam toward the inside. The seam of the pocket may then be closed. Baste it first, and close it with one stitching. The back edge of the pocket is caught to the front with bar stay-tacks. The upper edges are held by the waistband.

TROUSERS HAVING NO FLY CLOSING have the waistband divided into a front and back waistband, leaving an opening at each side of the trousers. In this case the extension on the side of the back pieces of the trousers is faced, thus forming the underlap for the opening. The loose edge of the pocket piece is then faced on both sides with the cloth, and two rows of stitching, a quarter of an inch apart close to the edge, give it a firm finish. Now the upper edges of the pocket are basted to the upper edge of the trousers front. (Ill. 369.)

Make a bar, overcast or buttonholed, between the two rows of stitching, catching through the cloth, and both sides of the pocket at the top and at the bottom of the opening.

The pocket may now be closed. Round off one or both of the corners, and, turning in the seam with the raw edges toward the inside of the pocket, stitch securely.

THE OUTSIDE SEAM of the trousers is closed next. In knickerbockers it is stitched in a seam all the way down. The lower edge of the leg is gathered in a casing with an elastic. In trousers finished with a band the extension allowed at the lower part for an opening at the side is turned under for a facing on the upper side, and faced and used as an extension on the under side. The band can be fastened with a buckle or with a button and buttonhole. After stitching the seam, turn the raw edges toward the front. From the outside, run a line of stitching one-eighth of an inch from the seam.

Now stitch and press open the inside seam of each leg. The two leg portions may then be joined, beginning the seam down the back at the waist, and

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text:
Collier's New Encyclopedia (1921)/Whiting, Lilian

WHITING, LILIAN, an American author; born in Niagara Falls, N. Y., Oct. 3, 1859. She was literary editor of the Boston “Traveler” in 1880-1890, and editor of the Boston “Budget” in 1890-1894. She then traveled in Europe. Her publications include: “The World Beautiful” (3 vols.); “From Dreamland Sent” (poems); “After Her Death”; “The Story of a Summer”; “A Study of the Life and Poetry of Elizabeth Barrett Browning”; “Kate Field: a Record”; “The Spiritual Significance”; etc.


text:
Translation:Ayil Meshulash/Discourse 7

This chapter contains sections 190 to 213

Section 190
To measure the stars of the heavens, in particular their degrees above the horizon at any hour, take for yourself a board and etch thereon a quarter circumference of a circle. Then take a piece of wood that is half the diameter of this circle and affix it by a peg to the center of the circle in such a way as the left end '''[WHY LEFT? ISN’T IT THE RIGHT SIDE?]''' of the wooden piece points toward the center. Affix it in such a way that it may be rotated to any point that you wish. Then attach to the left side of the wooden pointer two small pieces of wood. A hole should be made in each of these wood pieces so that one can see straight through both holes at once [in a line-of-sight]. Finally, along the circumference inscribe ninety degrees from point one to two, like so:

Section 191
If one then wishes to know the position of the sun, moon, or stars by degrees over the horizon, in any of the four directions of the world, wherever one may be, take this tool and stand in upon a flat surface that does not incline in any direction so that side CB is towards the ground and side AC is at a right angle over the plane of the earth. Then move the pointer, notated in the figure above as DC, until the desired star is seen through the holes. The left side of the pointer will then show at what point on the circumference is the star. For example, if the star is seen at point E, count the number of degrees from point B to point E. That amount is the number of degrees this star is above the horizon, because all circles, whether large or small, have the same system of degrees. Therefore, the degrees shown on the tool are the same as the degrees of the sphere of the sky right up to point A, which is directly overhead.

Section 192
To measure the height in degrees of the 'pivot of the world' [celestial pole], which is the point in the sky that does not move at all, that rotates around itself - if one knows the star that is in this place one can measure as was mentioned in the previous section. If one cannot locate this star there are several alternative methods.

Section 193
The first method: If you know the number of degrees that the sun is removed from the 'equalizing line' on that very day [this is the line encircling the globe under which the day and night are split equally in both the summer and winter, also known as the ecliptic], - which can be determined by knowing how many days it is from the beginning of the season - and then measure the height of the sun over the celestial horizon [the circle formed by the horizon], the sun's appearance within the southern constellations of Libra, Scorpio, Sagittarius, Capricorn, Aquarius and Pisces means that the distance in degrees between the sun and the ecliptic should be added to the height of the sun over the celestial horizon and that is the height of the ecliptic over the celestial horizon. The remaining degree measure between the ecliptic and 90 degrees is the height of the 'pivot', as is known.

If the sun is found in the northern constellations, which are Aries, Taurus, Gemini, Cancer, Leo and Virgo, subtract the distance of the sun from the ecliptic from its height over the celestial horizon and that equals the height of the ecliptic over the celestial horizon. The degree measure between that height of the ecliptic and 90 degrees is height of the pivot [the celestial pole].

If the sun is at the beginning of Aries or Scales [meaning it is found at the vernal or autumnal equinox, the points where the ecliptic and the celestial horizon meet] then the height of the sun is the height of the ecliptic.

Section 194
Method two: If through some calculation the distance between a star [or planet] and the celestial horizon is already known, whether to the north or south, the measurement of that star from the ecliptic together with the calculation in the previous section will yield the height of the pivot [or celestial pole]. Similarly, if the distance between a star and the pivot is known - if it is on the southern side of the pivot its distance from the pivot is subtracted from its height over the northern celestial horizon. If it is to the north of the pivot [celestial pole], add its distance from the pivot to its height over the celestial horizon and that is the height of the pivot [pole].

Section 195
Method three: Using the astrolabe [constructed in section 190], position it along the 'half-day horizon' [the celestial meridian], a circle running from the north up to the zenith, which is the top of the celestial sphere, and then down to the south. This line passes through the celestial pole.] Finding a star along this line south of the celestial pole, measure its height above the northern celestial horizon. Afterward, wait twelve hours until that star returns to a position along the celestial meridian, this time north of the celestial pole [as all stars in the sky rotate around the celestial pole once in twenty four hours]. Then measure its height over the celestial horizon. When the second height measurement is subtracted from the first, and the remainder is divided in half and either added to the second measurement or subtracted from the first, and the result is the height of the celestial pole over the celestial horizon.

Section 196
To determine the diameter of the sun [in degrees] measure the height over the celestial horizon of the top of the sun [on the celestial pole side] and measure the height over the celestial horizon of bottom edge of the sun [on the celestial horizon side]. The height of the top less the bottom is the [degree] measurement of the sun's diameter

Section 197
Another method is to measure the length of time it takes from the very start of the sun's rising over the land [in the morning] until its rising is complete. The proportion of the time that took to 24 hours [a whole day] will be the same proportion of 360 degrees to the degree measure of the sun's diameter.

The same would apply to the moon and other stars [planets] as well.

Section 198
In order to determine how many 'parsaos' [a land measurement, similar to miles, roughly 2.4–2.88 miles] the diameter or circumference of the Earth is, measure a star that is along the celestial meridian, calculating the distance of that star [or planet] from the zenith [overhead point] or from the celestial horizon. Following the celestial meridian along a direct north or south line, travel to another location and measure this star again. Subtract one measurement from the other. The remainder will be a portion of the degrees of the Earth’s circumference. Calculate the distance between the two points and set up the following proportion: The ratio of the [remainder] amount of degrees to a full 360 degrees is the same as the ratio of the distance between the two points and the Earth’s circumference. Once the circumference is known the diameter can be calculated based on the ratio of 22 to 7 [Pi, 3.14], as was written in section 77.

Section 199
Another method is to find a location where the sun is directly overhead at midday and then measure how many for ‘parsaos’ [north-south or east-west] there are no shadows cast. Afterward measure the diameter of the sun in degrees. The ratio of the diameter of the sun to 360 degrees is proportional to the ratio of the distance measured to the Earth’s circumference. Then the Earth’s diameter can also be calculated, as was seen previously.

Section 200
Another method is to use the eclipse of the sun or moon in two locations in a straight line, whereby the eclipse occurs in one location before the next. The ratio of the time between eclipses to 24 hours is the proportional to the ratio of the distance between the two points to the circumference of the Earth.

Section 201
Another method is to stand upon a very tall mountain, since the sunset at the mountaintop occurs later than on the plain. Measure how much longer it takes to reach sunset [on the mountain compared to the plain]. Then, use an astrolabe in the following position:

Side AB is parallel to the celestial horizon, and side AE is perpendicular to the celestial horizon. View the sun at point D, measuring how many degrees it is under the horizontal top of the astrolabe, which is angle BAC. Subtract the maximum sine from the secant of angle BAC. Then, set up the proportion: The ratio of the result of the subtraction to the maximum sine is proportional to the straight line height of the mountain to half of the Earth’s diameter. Once the diameter is known, the circumference can be calculated based on a ratio of 7 to 22 [Pi], as was shown previously.

Section 202
The proof to this requires a preface. How is it known that sunset occurs later on a mountain top than on a plain? The reason is that on a mountaintop the line of sight is able to extend to a lower angle, like so:

A man standing on the mountain AB over radius BE sees the sun at point D in the same manner as people at point C over radius CE. Therefore, the sunset on the mountaintop occurs at the same time as it does over the radius CE, which us later that when it occurs over radius BE. In so much as this forms angle BEC, and in triangle BEC the radius BE is also the maximum sine, line AC is the tangent of angle BEC, and line ABE is the secant of this angle, subtracting the maximum sine from the secant AE will yield line AB, which is the height of the mountain. If the height of the mountain and the length of the radius are known [and a ratio can then be determined,] it is possible to know how long after sunset on the plain will sunset on the mountaintop occur.[From Hebrew Commentary: Between the two points lies the secant of angle BEC. It can be determined from the secant, using the table of sines, how many degrees is the angle or arc of BC that is between sunset on the mountaintop and sunset on the plain.] Similarly, if one knows the length of the radius and the [sunset] time in the two locations the height of the mountain can be determined [From Hebrew Commentary: Through knowledge of the times one is able to determine other unknowns, as was seen in section 200. One who knows the measure of arc BC or angle BEG can determine, through use of the tables, how many units are needed to function as a secant for that angle, and after the measure of the maximum sine is determined, which is line BE, or half the diameter of the Earth, it can now be seen that the height of the mountain is the difference between the maximum sine and the secant line.]

Section 203
To measure the diameter of the moon’s orbit by its changing position: This method is based on the changing position of the moon relative to a point in the constellations that we see as we view it from the surface of the Earth. In this figure:

circle ABCD is the sphere of the constellations’ orbit, circle EFG is the sphere of the moon’s orbit, and circle HIJ is the Earth itself. Point K is the center of the Earth, and the moon stands at point F. The line extending from the earth’s center past the moon to the constellations, through the use of which these calculations will be based, is line KFB [the 'true position' of the moon relative to the Earth, not the 'apparent position' to the observer]. The line-of-sight line from an observer past the moon to the constellations is line HFC. The difference between them is angle HFK, as was noted in [section 143, per commentary]. Once you have determined A) angle HKF [via section 105, per Hebrew commentary], the angle of the moon's 'true position'; B) angle KHF, which is the supplementary angle of the angle of 'apparent position' [against the constellation from the viewer's perspective, which is angle AHC]; C) opposite angle HFK, which is the angle of distance [between points B and C is known since the previously mentioned angles are known]; D) and line HK, which is the radius of the Earth [known via section 198, per commentary], then you also know line HF, which is the distance from the observer to the moon, and line KF, which is the distance from the center of the Earth to the moon.

Section 204
After the distance from the center of the Earth to the moon has been determined, calculate how many degrees the moon currently stands in the circumference of its minor orbit. This is determined by the methods in section 150-153, whereby one determines all three angles HFC, HCF, and CHF in the figure in that section. [This figure matches the geocentric model, which views the moon as in larger orbit around the Earth as well as a minor orbit around the line of circumference of its orbit. This allows for calculation of the moon’s actual orbit, which is elliptical, rather than circular. Sometimes the moon is closer to Earth, sometimes it is farther away.] Recall that the length of line HF is already known, as the last section has shown, and HC, the line from the center of the Earth to the moon, can now be known, as can line CF [both of which end at point C on the circumference of the minor orbit]. Line CF is the radius the moon's minor orbit circle.

Section 205
Once the distance from the center of the Earth to the moon's minor orbit is known, the radius of the moon's orbit is known, whose center is removed from the center of the Earth [as stated, the center of the moon's orbit is not the center of the Earth], based on what was written in section 144. Then, after the length of the diameter of the moon's orbit [that is, the ‘virtual circular orbit’ based on the current distance of the moon from the Earth at the moment] is known, the circumference can be calculated, as has been shown previously. Once the circumference is known, one can calculate the diameter of the moon itself by setting up a proportion between the diameter of the moon [in degrees] and the circumference of its orbit, based on the degree measurement of the moon within the circumference of its orbit.

Section 206
To know the degree measure of the moon in its orbit use an astrolabe, as has been previously shown. After this one can calculate in relation to the Earth's center, as was shown in section 203, and then calculate in relation to the center of the moon's orbit, as has been shown.

Section 207
Determining the celestial meridian [IS THIS CORRECT?], upon which this entire calculation was based, requires the use of a magnet. It is a known property of a magnet to point to the north [and the meridian runs from north to south]. Similarly, if you know the location of the geographical pole this can also be determined, since it [the meridian] passes [in a line] directly overhead and through the geographical pole. [IS THIS CORRECT?]. Absent these, one may inscribe a circle on a board and place it flat, parallel to the celestial horizon. A peg is affixed to the center, and the place where the peg's shadow intersects with the circumference at sunrise and sunset is marked off. A line is drawn connecting these two points, and then this line is bisected by a perpendicular line. That line shows the direction of the celestial meridian [running north to south]. For example, in the accompanying figure:

A small peg is placed at point K. At sunrise, its shadow is runs along a line from K to between F and J, and at sunset from K to between points I and B. Line BF can be drawn across, and then line HKD is drawn perpendicular. That line represents the celestial meridian.

Section 208
Another method is to wait until before midday, and using the circle and the peg, observe the last moment before midday that the shadow touches a point on the circumference and note that point. Then, after midday, wait until the first instance of the shadow touching the circumference occurs, and note that point [NOT SURE OF THIS SECTION]. Connect these two points by a line, and the perpendicular line that bisects this line is the direction of the celestial meridian. For example, in the figure in the previous section, one may have seen that before midday the last point on the circumference touched by the shadow is E, and after midday the first point touched is C. With these two point the line CE is drawn and bisected by line HKD, as was done in the previous section, and this is the direction of the celestial meridian. If one desires he can make many concentric cycles around the center, each within the other [to get results closer to the moment of midday itself], observing when the shadow touches them before midday and after.

Section 209
If one desires to know whether it is before, after or exactly midday on any given day he may observe where the shadow falls across the inscribed circle [assuming the board it is inscribed on has not been moved after the previous observation]. If the shadow falls after line HKD it is before midday, if it falls before it is after midday. If it falls on the line itself it is midday. If one wishes to know the same information during the night he should use the same method with the moon provided the moon is in 'Nikud' [HEBREW TERM, COMMENTARY UNSURE OF TRANSLATION, MAY BE SYZYGY, WHEN THE MOON IS IN CONJUNCTION OR OPPOSITION, AS THAT SEEMS TO MATCH THE NEXT CLAUSE], or by the use of a planet or star if they are found in the sky opposite the sun 180 degrees.

Section 210
To know whether the season of ‘Nisan’ (spring) or ‘Tishrei’ (Fall) has arrived one need only to bisect line HKD with a perpendicular line, AKG [text has it as ‘line ABG’, but that does not seem correct], which would be parallel with lines BF and CE. The day that the shadow at sunrise or sunset reaches this line will be the [beginning of the] season of spring or fall.

Section 211
The closer the moon is to the Earth the greater the change of position in respect to the backdrop is when using the calculation method in section 203. This is because the proportion of line HK to line FK is equal to the proportion of the sine of angle HFK to the sine of angle KHF. Since this is so, and reduction in the length of line FK will enlarge angle HFK.

Section 212
Similarly, the closer the moon is to the celestial horizon the greater its apparent change of position will be. This is because the proportion of the lines to each other is equivalent to the proportion of the angle’s sines to each other, as was written in the previous section. As the width of angle HKF decreases its sine will increase, as was noted in section 82. Therefore, the closer it is to the celestial horizon the larger the sine of angle KHF will be, and the larger angle HFK will be. The result is that at the horizon the change in perception will be at its greatest while overhead there will be no perceptible change at all.

Section 213
In order to determine this change at every angle possible from overhead to the horizon one must know the distance of the moon from the center of the Earth, which is line KF; the radius of the earth, line HK; the moon’s distance in degrees from overhead, which is angle HK, or the complementary angle, angle KHF. These values allow one to calculate the remaining necessary values.

איל משולש

title: Translation:Ayil Meshulash/Discourse 7


text:
Page:Newdressmakerwit00butt.djvu/23

CHAPTER 3 ALTERING THE LENGTH OF PATTERNS Lengthening or Shortening a Waist Pattern — Lengthening and Shortening Sleeve Patterns Increasing or Decreasing the Size of Collars — Lengthening and Shortening Skirt Patterns UTTERICK PATTERNS are made with such expert skill and are the result of such scientific accuracy and study that they are an absolutely perfect fit for the woman of average proportions. But it often happens that a woman varies from the normal at some point. The variation may not be marked enough to be noticeable. But if she is an inch or two long or short waisted, if she is slightly round-shouldered, if the upper part of her arm is long in proportion to the lower part, her patterns wall have to be altered a trifle to make them correspond to her figure. The alterations are extremely easy to make, but this point again illustrates the advantage of making your clothes to suit your figure, to fit the irregularities of the figure that are taken care of in this chapter.

The changes that are given here may be made without altering the original lines of the pattern, though they should be made in the pattern and not in the garment after it is cut out. If you attempt to make these changes in fitting the garment it will mean more work than if you had made a simple alteration in the pattern before cutting out your material. Trying to alter the garment itself is unsatisfactory, even with the additional work, for these changes can not be made after the garment is cut.

Almost every woman knows if there is any variation from normal in her figure. She knows if she is tall or short, long-waisted or short-waisted, if she has round shoulders, a narrow back, a flat chest or a broad back, etc. Before cutting your material you must be sure that the pattern is the right length and shape for your particular figure.

ADAPTING PATTERNS TO LONG OR SHORT WAISTED FIGURES—Before cutting your material have your figure measured from the collar seam at the back of the neck to the normal waistline, and close up under the arms to the waistline. Make a memorandum of these measurements as they are taken.

Now take your lining pattern and take the corresponding measurements of the pattern. Remember that the back pattern has a ⅜ of an inch seam allowance at the neck and that there is a ⅜ of an inch seam allowance at the armhole edge under the arm.

Compare your measurements with the measurements of the pattern. In most cases it is sufficient to alter the waist length at the lower part. Some figures, however, are long-waisted from under the arm to the waistline and short from under the arm to the neck. If this is your case you will discover it in comparing your measurements with the measurements of the pattern. If the lower part is too long lay a plait across each piece about 2½ inches above the waistline. (Ill. 27.) The 19

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text: Popular Science Monthly/Volume 67/August 1905/The Value of Old Age


text:
Collier's New Encyclopedia (1921)/Whitlock, Brand

WHITLOCK, BRAND, an American writer and diplomat; born in Urbana, O., in 1869. After a common school education he entered the field of journalism in Toledo, O., but, in 1890, went to Chicago to join the staff of the Chicago “Herald.” Three years later he became a clerk in the office of the secretary of state of Illinois. During this period he studied law and was admitted to the bar of Illinois in 1894 and, in 1897, to the bar of Ohio. He began to practice law in Toledo, where he was elected mayor in 1905, being re-elected to the same office for three other terms. Under his administration a new charter was granted the city and such innovations as the initiative, the referendum, the recall of public officials, and direct nominations were instituted. Meanwhile he was also becoming known as a regular contributor to major American magazines, both as a writer of verse and of

prose articles. In 1913 he was appointed Minister to Belgium, where he distinguished himself for his expert handling of the situation during the World War. War having been declared against Germany by the United States in April, 1917, Mr. Whitlock was recalled, but after the signing of the armistice and the restoration of the Belgian Government, he returned to Brussels as Ambassador. Among his many works are: “Her Infinite Variety” (1904); “The Gold. Brick” (1910); “Belgium, a Personal Narrative” (1919).


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20 depth of the plait will depend on the amount of difference between your figure and the length of the pattern.

If you find that your underarm measure and that of the pattern are the same, but that you are shorter-waisted than the pattern, the difference is in the upper part. Decrease the length of the upper part of the pattern by laying a plait across the front and back, half-way between the under arm and the neck. (Ill. 27.)

If you are long-waisted instead of short-waisted the lining pattern can be altered at the same points, by cutting the pieces of the pattern and separating them sufficiently to give you the length you need. (Ill. 28.)

In laying plaits across the pattern the edge of the pattern should be evened off after the plaits are laid.

If an alteration is made in the length of the lining a corresponding alteration must be made in the pattern of the outside of the waist or dress, taking out or putting in the same amount and in the same places.

LENGTHENING OR SHORTENING THE TWO SEAM SLEEVE PATTERNS—Your arm may measure exactly the same as the sleeve pattern, but its length might be made up of a long upper arm and of a short forearm. In that case the elbow of the pattern would not come in the right place on your arm. Or your arm may be perfectly proportioned, but it may be longer or shorter than the average, in which case an alteration would have to be made in both the upper and lower part of the sleeve pattern to make it the right length for you.

For a two-seam sleeve pattern measure along the inside of your arm from the armhole to the bend of the elbow, and from the bend of the elbow to the wrist. You will need both these measurements, so as to be sure that the sleeve elbow will come in the right place. Measure your sleeve pattern along the inside seam of the upper part from ⅜ of an inch below the armhole edge to the small double perforations which indicate the elbow. Measure from the elbow perforations to within ⅜ of an inch of the wrist edge. The ⅜ of an inch on each edge is the seam finish.

Compare your arm measurements with the measurements of the pattern. If the pattern is too long for you above the elbow lay a plait across the pattern a little above the elbow. (Ill. 29.) If there is an inch difference in length the plait should be ½ of an inch deep. If the lower part of the sleeve is too long for you lay a plait across the lower part below the elbow. (Ill. 29.)

The same alterations should be made in the under part of the sleeve pattern, making the plaits the same depth that you made in the upper part of the pattern.

When the plaits are laid the edge of the pattern will become uneven. If the arm is very full the space between the elbow and the greater width of the upper part of the sleeve should be filled out, giving you an even line. If the arm is not large you can trim off the extra width and get your even outline in that way.

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text: Popular Science Monthly/Volume 67/August 1905/A Suggestive Case of Nerve-Anastomosis


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text: Popular Science Monthly/Volume 67/August 1905/A Visit to Luther Burbank


text: Popular Science Monthly/Volume 67/August 1905/Some Phases of the Educational Problems in China


text: Popular Science Monthly/Volume 67/August 1905/The Social Phase of Agricultural Education


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text: Popular Science Monthly/Volume 67/August 1905/Education for Efficiency


text: Popular Science Monthly/Volume 67/August 1905/Address of President Roosevelt Before the National Educational Association


text: Popular Science Monthly/Volume 67/August 1905/The Progress of Science


text: Popular Science Monthly/Volume 67/September 1905/China's Renaissance


text: Popular Science Monthly/Volume 67/September 1905/The Science of Plant Pathology


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text: Popular Science Monthly/Volume 67/September 1905/Sleep and its Regulation


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Collier's New Encyclopedia (1921)/Whitman, Charles Seymour

WHITMAN, CHARLES SEYMOUR, an American public official, born at Norwich, Conn., in 1868. He graduated from Amherst College in 1890. After admission to the bar, he practiced in New York City. From 1901 to 1903 he was assistant corporation counsel, and from 1904 to 1907 was a city magistrate. He was appointed judge of the Court of Special Sessions in 1907, and from 1910 to 1914 was district attorney of New York county. In 1914 he was elected governor of New York. He was re-elected in 1916.


text: Popular Science Monthly/Volume 67/September 1905/State University Salaries


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text: Popular Science Monthly/Volume 67/September 1905/Nature and Man


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text: Popular Science Monthly/Volume 67/September 1905/General Education for Engineers


text: Popular Science Monthly/Volume 67/September 1905/Quackery


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Rh

If the sleeve pattern requires lengthening make the alteration at the same places in the pattern, cutting across the pattern and separating the pieces as in Illustration 30. Correct the outline of the edge in the same way as when the sleeve is shortened.

LENGTHENING OR SHORTENING A ONE-SEAM SLEEVE PATTERN OR A KIMONO SLEEVE PATTERN—One-seam sleeves and kimono sleeves fit the figure quite differently from the two-seam sleeve and so have to be measured in a different way. In using either of these sleeves have your arm measure taken from the center of your back along the outside of your arm to the wrist with the arm bent. (Ill. 31.)

In the case of a one-seam sleeve pin the sleeve pattern to the back pattern with the armhole edges lapping ¾ of an inch. If there is a cuff pin it to the lower part of the sleeve lapping them ¾ of an inch. Turn up the cuff in case the cuff is double. (Ill. 32.) Compare your measurements with the measurements of the pattern, and if the pattern is long or short for you, alter it as shown in Illustrations 33 and 34. A sleeve of this kind has no elbow curve, so that it is

not necessary to alter it above and below the elbow. It is simply a question of adding or subtracting length at the correct point. If the pattern is an inch too long for you lay a ½-inch plait across the pattern. (Ill. 33.) If it is too short slash the pattern and separate the pieces. (Ill. 34.) If your arm is full you can correct the outline by filling in below the plaits and above the slash. If it is of average size you can trim down above the plait and below the slash.

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text: Category:Transportation

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text: Popular Science Monthly/Volume 67/September 1905/How Canada Is Solving her Transportation Problem


text: Popular Science Monthly/Volume 67/September 1905/The Ancestors of the Big Trees


text: Popular Science Monthly/Volume 67/September 1905/The Progress of Science


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22 {| align=center
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 * Ill. 36. Increasing the Size of Collars and Neckbands ||| Ill. 37. Decreasing the Size of Collars and Neckbands
 * }
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In the kimono sleeve pattern measure from the center of the back below the neck to within ⅜ of an inch of the lower edge of the sleeve. (Ill. 35.) This ⅜ of an inch is a seam finish. The alterations for lengthening or shortening a kimono sleeve are exactly the same as for the one-seam sleeve. (Ills. 33 and 34.)

INCREASING OR DECREASING THE SIZE OF THE NECKBAND AND COLLAR—If the neckband of a pattern is not the right size for your neck it should be altered at the center back. If it is too small cut it through the center and separate the pieces sufficiently to make it the right size for you. Illustration 36 shows this alteration for neck-band and a turndown collar.

If the collar pattern is too large for you lay a plait across the center back. (Ill. 37.)

LENGTHENING AND SHORTENING GORED SKIRTS—Have your measure taken from the normal waistline at the center-front to the distance from the floor at which you wear your skirts. This length varies with different fashions. Add an inch to your measurement for the give and take of making the skirt. The front-length measurement of your skirt pattern will be given on the pattern envelope. Compare your measure with the length of the pattern. If the pattern is too long for you, lay a plait straight across each gore about six inches below the hipline. (Ill. 38.) The hipline is seven inches below the normal waistline. (Ill. 38.)

If the gores are cut with one straight edge, measure the plait at the straight edge. If both edges of the gore are bias, measure the plait at the line of large double perforations that mark the straight thread of the material.

If the figure is full, the edge of the gore on the bias side should be filled out from the folded plait to the

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text: Popular Science Monthly/Volume 67/October 1905/The Lapses of Consciousness


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text: Popular Science Monthly/Volume 67/October 1905/Fake Weather Forecasts


text: Popular Science Monthly/Volume 67/October 1905/The College of the White Deer Grotto


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text: Popular Science Monthly/Volume 67/October 1905/Unconscious Assumptions in Economics


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Rh hip. (Ill. 38.) If the figure is slight, this extension which comes from the plait can be trimmed off.

To lengthen a gored skirt pattern, cut across each gore six inches below the hipline and separate the pieces as much as necessary. (Ill. 39.)

In very rare cases it is sometimes necessary to alter the length of a skirt pattern as much as four or five inches. In an extreme case of this kind it is better to make half the alteration six inches below the hip as shown in the Illustrations 38 and 39, and the other half at the lower edge of the pattern, turning up the lower edge to shorten it and adding to it to lengthen it.

In a circular skirt the pattern should be altered at the lower edge.

LENGTHENING OR SHORTENING A STRAIGHT SKIRT PATTERN—The length of a straight skirt can be altered at the bottom, turning up the lower edge if it is too long and adding more length in cutting if it is short.

DRAPED SKIRT PATTERNS, TUNIC SKIRT PATTERNS, ETC.—If it is necessary to alter the length of a draped skirt pattern or a tunic skirt pattern, or a pattern with tucks or marks for trimmings, alter it according to the instructions given on the pattern envelope. The position for the alteration will vary with the skirt.

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Collier's New Encyclopedia (1921)/Whitman, Marcus

WHITMAN, MARCUS, an American pioneer; born in Rushville, N. Y., Sept. 4, 1802. In 1836 he emigrated with a number of others to work as a missionary among the Indians of the Upper Columbia. Accompanied by his young wife he crossed the plains by wagon, being the first person to reach the Pacific by this means. He was soon followed by a large number of emigrants who settled in what was then known as Oregon, and which now forms the States of Oregon, Washington, and Idaho. At this time the Hudson's Bay company were using every possible means to secure this territory to the English. When this plan became evident to Dr. Whitman he decided to take every precaution to forestall it. The Ashburton-Webster treaty was then before Congress and was expected to settle the Oregon question. Knowing that the Government should have full information as to the true state of affairs, Whitman rode over 3,000 miles on horseback, enduring all the hardships of a Western winter in the mountains, and reaching Washington on March 3, 1843, only to find that the treaty had been signed. Fortunately the Oregon question had not been included, Dr. Whitman at once went to work and taught the Government the value of the land it had deemed worthless, demonstrated to the people the fertility of the soil of Oregon, and the fact that it could be reached by wagon, and then returned at the head of 1,000 emigrants. By his

daring ride and his earnest endeavors Dr. Whitman won this great section for the United States, and the results of his work were secured by the treaty of 1846. In 1847, with his wife and some others, he was massacred by the Cayuse Indians.


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text: Popular Science Monthly/Volume 67/October 1905/The Distribution of the Daily Time of Cornell Students


text: Popular Science Monthly/Volume 67/October 1905/Greek Ideas of Vulcanism


text: Popular Science Monthly/Volume 67/October 1905/Western Explorations for Fossil Vertebrates


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Collier's New Encyclopedia (1921)/Whitman, Sarah Helen Power

WHITMAN, SARAH HELEN POWER, an American poet; born in Providence, R. I., in 1803. She married John W. Whitman, a Boston lawyer; was once engaged to Edgar Allan Poe, afterward writing a defense of him entitled “Edgar A. Poe and his Critics” (1860); and was noted for her conversational powers. She published several volumes of poems, among them being the volume “Hours of Life, and Other Poems” (1853); also “Fairy Ballads,” written with her sister, Anna M. Power. She died in Providence, R. I., June 27, 1878.


text: Popular Science Monthly/Volume 67/October 1905/The Progress of Science


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text: Popular Science Monthly/Volume 67/November 1905/The Botanical Garden at Buitenzorg, Java


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 AND THEIR WORK

From the Fifteenth Century to the period of the Great Rebellion, An unwritten chapter in the History of the Organ. A Lecture delivered before the College of Organists, November 15, 1864.

BY , Ll.d. Member of the Royal Academy of Music in Stockholm, Musical Examiner in the Royal College of Preceptors, etc.

LONDON: WM. REEVES. 83 CHARING CROSS ROAD, W.C.2.


text: Popular Science Monthly/Volume 67/November 1905/Hypnotism, its History, Nature and Use


text: Popular Science Monthly/Volume 67/November 1905/Physicians and Philosophers


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text: Popular Science Monthly/Volume 67/November 1905/Soil Fertility