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Because of the foregoing proportions, we have 'DG+BE OmxCF
DG-BE , and Du
OC Dmx FO
2 Om x CF ; and therefore DG+BEOC
OC 2DmxFO and DG — BE =
Hence, if the mean arch AC be supposed that of 60°; then OF being the co-line of 60°, = sine 30o = chord of 60° = LOC, it is manifest that DG BE will, in this case, be barely = Dm; and consequently DG = Dm + BE. From whence, and the preceding corollary, we have theie two useful theorems.
1. If the fine of the mean, of three equidifferent arches (supposing radius unity) be multiplied by twice the co-fine of the common difference, and the yine of either extreme be subtracted from the product, she remainder will be the fine of the other extreme.
2. The fine of any arcb, above 60 degrees, is equal to the fine of another arch, as much below 60°, together with the fine of its excess above too.
taken in a geometrical sense, de
OC notes a fourth-proportional to OC, Om and CF; out, arithmetically, it fignifies the quantity arising by dividing the product of the measures of Om and CF by that of Od. UnderLand the like of others.
To find the fine of a very small arch; suppose that of 15'
It is found, in p. 181. of the Elements, that the length of the chord of 75% of the femi-periphery is expressed by ,00818121 (radius being unity); therefore, as the chords of very small arches are to each other nearly as the arches themselves (vid. P. 181.) we shall have, as : 567:: 900818121 : ,008726624, the chord of 350, or half a degree ; whose 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 sine of I'be required, it will be, 15':1':: ,004363312 : 9000290888, the sine 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 same degree of exactness), then let the operations, in p. 181, be continued to II bisections, and a greater number of decimals be taken; by which means you will get the chord of city part of the ferni-periphery to what accuracy you please : then, by proceeding as above (for finding the sine of 15'), the fine of 1 minute will also be obtained to a very great degree of exactness.
PROP. IV. To hew the manner of construeling the trigonomea. trical 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' line o' = sine 2'.
And thus are the fines of 6', 7, 8', &c. fuccessively derived from each ocher.
The lines of every degree and minute, up to 60°, being thus found; those of above 60° will be had by addition only (from Theor. 2. 2. 15.) then, the fines being all known, the tangents and secants will likewise 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 first lix places true in each number; which is suffi. ciently exact for all common purposes.
SCHOLIUM. Although what has been hitherto laid down for constructing the trigonometrical-canon, is abundantly sufficient for that purpose, and is also very easily demonstrated; yet, as the first sine, 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 tolerably 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-line of 15', which is given (by p. 181 of the Elements) = *V2 tv 3 =,965925826, &c. (=the supplement chord cf 30°) and the fine of 18°, which is = IV-=
2309017, &c. (equal to half the side of a decagon inscribed in the circle) let the co-fine of 3°, the difference between 186 and 15', be found *, from which the co-line of 45' will be had, by two bisections only: whence the lines of all the arches in the progression 1° 30', 2° 15°, 3° 00', 3° 45', &c. may be determined (by Theor. 1. p. 15.) and that to any assigned 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 thefe, the fines of all the intermediate arches, to every single minute.
This, if you desire 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 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 feconds, and even to thirds, in most cases), then the operation may be as follows:
1°. Multiply the sum of the fines 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 sines) by the fraction ,0000000423, for a first product; and this, again, by 22, for a second product; to which last, let's of the difference of the two proposed sines (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 (fuppofing radius unity), is found by adding the product of their fines to that of their co-fines; as is hereafter demonstrated.
tremes, will be ,0002904915, the excels of the
2°. 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 excess; and, to the sum, add the first remainder ; to this sum 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, let it be proposed to find the lines of all the intermediate arches between 3° 00' and and 3° 45' to every single minute, those of the extremes being given, from the foregoing method, equal to 905233595 and ,06540312 respectively. 'Here, the sum of the lines of the extremes being muldiplied by ,0000000423, the first product will be ,00000000498, &c. or ,0000000050, nearly (which is sufficiently exact for the present purpose); and this, again, multiplied by 22, gives ; 00000011 for a 2d product; which added to ,0002903815, t's part of the difference of the two given ex
fine of 3° or above that of 3° oo'.
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.