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To find the fine of a very small arch; JURPOSE that of 15'.
It is found, in p: 181. of the Elements, that the length of the chord of $# of the semi-periphery, is expreffed by ,00818121 (radius being unity); therefore, as the chords of very finall arches are to each other nearly as the arches themselves (vid. p. 181.) we shall have, as ito::,00818121: ,008726624, the chord of to, or half a degree ; whose half, or ,004363312, iş therefore the fine of 15', very nearly.
From whence the fine of any inferior arch may be found by bare proportion. Thus, if the sine of s' be required, it will be, 15":11: : ,004363312; ,CC0290888, the line of the arch of one minute, nearly.
But if you would have the fine of more exactly determined (from which the fines of other arches may be derived with the same degree of exactness); then let the operations, in p. 181, be continued to 11 bifections, and a greater number çf decimals be taken; by which ineans you will get the chord of the part of the semi-periphery to what accuracy you please: then, by proceeding as above (for finding the line of 15), the sine of 1 minute will also be obrained to a very great degree of exactness.
PROP. IV: 1. To fhew the manner of confrutting the trigonometrical canon.
First, find the fine of an arch of one minute, by the preceding Prop. and then its co-fine, by
Prop. 1. which let be denoted by C; then (by
2C x sine 1' sine o'=fine 2'.
And thus are the fines of 6,7', 8', &c. fuccessively derived from each other.
The fines of every degree and minute, up to 60°, being thus found; those 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 likewise become known, by Prop. 1.
Note, If the fine of every sth minute, only, be computed according to the foregoing method, the lines of all the intermediate arches may be had from thence, by barely taking the proportional parts of the differences, and that so near as to give the first six places true in each number; which is sufficiently exact for all common purposes.
SCHOLIUM. Although what has been hitherto laid down for constructing the trigonometrical-canon, is abundantly fufficient for that purpose, and is also very easily demonstrated; yet, as the first fine, from whence the rest are all derived, must be carried on to a great number of places, to render the numerous deductions from it but colerably exact (because in every operation the error is multiplied), I shall here subjoin a different method, which will be found to have the advantage, not only in that, but in many other respects.
First, then, from the co-fine of 15°, which is given (by p. 181 of the Elements) = įv 2+1
3 =965925826, &c. (=the supplement chord of 30°) and the line of 18°, which is =VI-=
,309017, &c. (equal lo half the side of a decagon inscribed 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 bisections only: whence the lines 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 lines of all the terms of the progression 45', 1° 30', 2° 15', &c. up to 60°, being thus derived, the next thing is to find, by help of these, the lines of all the intermediate arches, to every single minute.
This, if you desire no more than the 4 or 5 firla 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 insisted on, and you would have a table carried on to 7 or 8 places, each number (which is sufficient to give the value of an angle to seconds, and even to thirds, in most cases) then the operation may be as follows:
1°. Multiply the sum of the lines of any two adjacent terms of the progression 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 first product; and this, again, by 22, for a second product; to which last, let it's of the difference of the two proposed fines (or extremes) be added, and the sum will be the excess of the first of the intermediate sines above the leffer extreme.
• Note, The co-fine of the difference of two arches (supposing radius unity), is found by adding the produ&t of their fines to tbat of their co-fines; as is hereafter demonstrated. 3
2o. From this excess let the first product be continually subtracted; that is, first, from the excess itself; then from the remainder ; then from the last remainder, and so on 44 times.
3°. To the leffer extreme add the forementioned excefs; and, to the sum, add the first remainder; to this sumn add the next remainder, and so on continually: then the several sums thus arising will respectively exhibit the fines of all the intermediate arches, to every single minute, exclusive of the last; which, if the work be right, will agree with the greater extreme itself, and therefore will be of use in proving the operation.
But to illustrate the matter more clearly, lat it be proposed to find the fines of all the intermediate arches between 30 oo' and 3° 45' to every single minute, those of the extremes being given, from the foregoing method, equal to ,05233595 and ,06540312 respectively. Here, the sum of the lines of the extremes being multiplied by ,0000000423, the first product will be ,00000000498, &c. or ,00000000 50, nearly (which is sufficiently exact for the present purpose); and this, again, multiplied by 22, gives ,000000 II for a 2d product; which added to ,C002903815,
part of the difference of the two given extremes, will be ,co02904915, the excess of the fine of 30 or above that of 3° oo'. From whence, by proceeding according to the 2d and 3d rules, che fines of all the other intermediate arches are had, by addition and subtraction only. See the operation.
4865 1f rem. ,0526264415 fine 3° 1 50
2904865 4815 24 rem. ,0529169280 fine 302 50
4765 34 rem. ,0532074095 fine 3° 3' 50
4715 4 rem. ,0534978860 fine 304 50
4665 5th rem. ,0537883575 sine 3° 5' 50
4615 6ih rem. ,0540788240 sine 306 50
2904615 4565 7th rem. ,0543692855 line 307 50
4515 8th rem. ,0546597420 sine 3° 8' 50
4465 9th rem. ,0549501935 sine 39 50
4415 10th rem. ,0552406400 fine 3° 10 &c.
Again, as a second example, let it be required to find the lines of all the arches, to every minute, between 59° 15' and 60° oo'; those of the two extremes being first found, by the preceding