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COROLLARY I.

Because of the foregoing proportions, we have DG+BE OmxCF

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*
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and Dv

2

OC

(DGBE)

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Hence, if the mean arch AC be supposed that of 60°; then OF being the co-fine of 60°, fine 30° chord of 60° OC, it is manifeft that DG BE will, in this cafe, be barely Dm; and confequently DG = Dm + BE. From whence, and the preceding corollary, we have thele two useful theorems.

1. If the fine of the mean, of three equidifferent arches (Suppofing radius unity) be multiplied by twice the co-fine of the common difference, and the jing of either extreme be fubtracted from the product, the remainder will be the fine of the other extreme.

2. The fine of any arch, above 60 degrees, is equal to the fine of another arch, as much below 60°, together with the fine of its excefs above 60°.

*Note,

Om X CF

ос

taken in a geometrical fenfe, de

notes a fourth-proportional to OC, Om and CF; but, arithmetically, it fignifies the quantity arifing by dividing the product of the measures of Om and CF by that of OC. Underftand the like of others.

PROP.

PROP. III.

To find the fine of a very small arch; fuppofe that of 15'.

It is found, in p. 181. of the Elements, that the length of the chord of of the femi-periphery उठक is expreffed by ,00818121 (radius being unity); therefore, as the chords of very fmall arches are to each other nearly as the arches themselves (vid. p. 181.) we shall have, as :::,00818121: ,008726624, the chord of, or half a degree; whofe half, or ,004363312, is therefore the fine of 15', very nearly.

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From whence the fine of any inferior arch may be found by bare proportion. Thus, if the fine of 1' be required, it will be, 15': 1'::,004363312: ,000290888, the fine of the arch of one minute, nearly.

But if you would have the fine of 1' more exactly determined (from which the fines of other arches may be derived with the fame degree of exactness); then let the operations, in p. 181, be continued to II bifections, and a greater number of decimals be taken; by which means you will get the chord of part of the femi-periphery to what accuracy you pleafe: then, by proceeding as above (for finding the fine of 15′), the fine of 1 minute will alfo be obtained to a very great degree of exactness.

PROP. IV.

To fhew the manner of constructing the trigonometrical canon.

Firft, find the fine of an arch of one minute, by the preceding Prop. and then its co-fine, by Prop.

Prop. 1. which let be denoted by C; then (by Theor. I. p. 13.) we fhall have

'

2C x fine fine o'

2C X fine 2 fine 'fine 3'.

fine 2′.

2C x fine.3'
2C x fine 4'

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fine 2'
fine 3'

fine 4'.

fine 5'.

And thus are the fines of 6', 7, 8', &c. fucceffively derived from each other.

The fines of every degree and minute, up to 60°, being thus found; thofe of above 60° will be had by addition only (from Theor. 2. p. 15.) then, the fines being all known, the tangents and fecants will likewife become known, by Prop. 1.

Note, If the fine of every 5th minute, only, be computed according to the foregoing method, the fines of all the intermediate arches may be had from thence, by barely taking the proportional parts of the differences, and that fo near as to give the first fix places true in each number; which is fufficiently exact for all common purposes.

SCHOLIUM.

Although what has been hitherto laid down for, conftructing the trigonometrical-canon, is abundantly fufficient for that purpofe, and is alfo very eafily demonftrated; yet, as the first fine, from whence the reft are all derived, must be carried on to a great number of places, to render the numerous deductions from it but tolerably exact (because in every operation the error is multiplied), I fhall here fubjoin a different method, which will be found to have the advantage, not only in that, but in many other refpects.

First, then, from the co-fine of 15°, which is given (by p. 181 of the Elements) = ± √ 2 + √ 3 =,965925826, &c. (= the fupplement chord of 30°) and the fine of 18°, which is÷√ ÷ = C ,309017,

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,309017, &c. (equal to half the fide of a decagon infcribed in the circle) let the co-fine of 3°, the difference between 18° and 15°, be found *; from which the co-fine of 45′ will be had, by two bifections only: whence the fines of all the arches in the progreffion 1° 30′, 2° 15°, 3° 00′, 3° 45′, &c. may be determined (by Theor. 1. p. 15.) and that to any affigned degree of exactness.

The fines of all the terms of the progreffion 45′, 1° 30′, 2° 15', &c. up to 60°, being thus derived, the next thing is to find, by help of thefe, the fines of all the intermediate arches, to every fingle minute.

This, if you defire no more than the 4 or 5 first places of each (which is exact enough where nothing less than degrees and minutes is regarded), may be effected by barely taking the proportional parts of the differences.

But if a greater degree of accuracy be infifted on, and you would have a table carried on to 7 or 8 places, each number (which is fufficient to give the value of an angle to feconds, and even to thirds, in moft cafes), then the operation may be as follows:

1o. Multiply the fum of the fines of any two adjacent terms of the progreffion 45', 1° 30', 2° 15′,3° 00′,3° 45', &c. (betwixt which you would find all the intermediate fines) by the fraction ,0000000423, for a firft product; and this, again, by 22, for a fecond product; to which laft, let 75 of the difference of the two propofed fines (or extremes) be added, and the sum will be the excess of the firft of the intermediate fines above the leffer extreme.

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Note, The co-fine of the difference of two arches (fuppofing radius unity), is found by adding the product of their fines to that of their co-fines; as is hereafter demonflrated.

2o. From

.

2°. From this excess let the first product be continually fubtracted; that is, firft, from the excess itself; then from the remainder; then from the laft remainder, and fo on 44 times.

3°. To the leffer extreme add the forementioned excefs; and, to the fum, add the first remainder; to this fum add the next remainder, and fo on continually then the feveral fums thus arifing will refpectively exhibit the fines of all the intermediate arches, to every single minute, exclufive of the laft; which, if the work be right, will agree with the greater extreme itself, and therefore will be of ufe in proving the operation.

But to illuftrate the matter more clearly, let it be proposed to find the fines of all the intermediate arches between 3° 00' and and 3° 45′ to every single minute, thofe of the extremes being given, from the foregoing method, equal to 05233595 and 06540312 refpectively. Here, the fum of the fines of the extremes being multiplied by ,0000000423, the firft product will be ,00000000498, &c. or,0000000050, nearly (which is fufficiently exact for the prefent purpose); and this, again, multiplied by 22, gives ,00000011 for a 2d product; which added to ,0002903815,

I

part of the difference of the two given extremes, will be ,0002904915, the excels of the fine of 3° 01' above that of 3° 00'. From whence, by proceeding according to the 2d and 3d rules, the fines of all the other intermediate arches are had, by addition and fubtraction only. See the operation.

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