1 PROP. III. To find the fine of a very small arch; suppose that of 15. 384 It is found, in p. 181. of the Elements, that the length of the chord of of the femi-periphery is expreffed by ,00818121 (radius being unity); therefore, as the chords of very finall arches are to each other nearly as the arches themfelves (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. 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: cc0290888, 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 exactnefs); then let the operations, in p. 181, be continued to 11 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 please: 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 exactnefs. I I PROP. IV. To fhew the manner of conftructing the trigonometrical canon. Firft, find the fine of an arch of one minute, by the preceding Prop. and then its co-fine, by X Prop. Prop. 1. which let be denoted by C; then (by 2C X fine 1'-fine o-fine 2'. 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 so near as to give the firft 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 purpose, and is also very eafily demonftrated; 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 nume rous 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. Firft, 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== ,309017, C ,309017, &c. (equal to half the fide of a decagon inscribed in the circle) let the co-fine of 3°, the difference between 18° and 15o, 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 these, 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 infisted 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 seconds, and even to thirds, in moft cafes) then the operation may be as follows: 1. 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 first product; and this, again, by 22, for a fecond product; to which laft, let's of the difference of the two proposed fines (or extremes) be added, and the fum will be the excefs of the first of the intermediate fines above the leffer extreme. *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 demonstrated. 3 2o. From 2°. From this excefs let the firft product be continually fubtracted; that is, firft, from the excess itself; then from the remainder; then from the last remainder, and fo on 44 times. 3°. To the leffer extreme add the forementioned excess; 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 use in proving the operation. But to illuftrate the matter more clearly, lat it be propofed to find the fines of all the intermediate arches between 3° 00′ 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,co02903815, part of the difference of the two given extremes, will be ,co02904915, the excefs of the fine of 3° or 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 subtraction only. See the operation. ng at m € 2 ,000290 ,000290 4915 excess ,0000000050 ,05233595 fine 3° 0' ft 4865 1 rem. 0526264415 fine 3° 1′ 50 2904865 4815 2 rem. 0529169280 fine 3° 2′ 4765 3d rem. 0532074095 fine 3° 3′ 4715 4th rem. 0534978860 fine 3° 4′ 4665 5th rem. 0537883575 fine 3°5′ 50 2904665 4615 6th rem. 0540788240 fine 3° 6' 4565 7th rem. 0543692855 fine 3° 7′ 50 2904565 4515 8th rem. 0546597420 fine 3°8′ 4465 9th rem. 0549501935 fine 3° 9′ 4415 10th rem. 0552406400 fine 3° 10 &c. 1 &c. Again, as a fecond example, let it be required. to find the fines of all the arches, to every minute, between 59° 15′ and 60° 00'; thofe of the two extremes being first found, by the preceding method. |