sine, 6: 6+s2::s: a, and hence as

since a and s are very small, a3 will approach extremely near to s3, and it may, therefore, be inferred conversely,

A convenient approximation for the versed sine of an arc is easily derived from the fundamental property of the lines themselves; for 2AO.AD AB BD'+AD2, or employing to denote the versed sine, 2v=s+v2, and s2 72

=

+ If, therefore, the arc be small, it may be suffi

2

-;

ciently near the truth to assume ; but should greater accuracy be required, substitute this value of v in the 54 second term of the complete expression, and v + which will form a very close approximation.

2 8

Calculation of the Trigonometric Lines.

The preceding theorems contain all the principles required in constructing Trigonometric Tables. The radius being denoted by unit, the several lines connected with the circle are referred to that standard, and are generally computed to seven decimal places.

The first object is to compute the SINES for every arc of the quadrant.

Since the semicircumference of a circle whose radius is unit was found, by the scholium to Prop. 30. Book VI. of the Elements, to be 3.1415926536, the length of the arc of

1

one minute is .0002909, which, in so small an arc, may be assumed as equal to the sine, and consequently the versed sine of a minute (.0002909) = .000,000,042,308. Whence, by cor. 3. to Prop. 3. sin(A + 1')= 2sinA— 2sinA X.000,000,042,308-sin(A-1'); and therefore, by a series of repeated operations, the intermediate arc being successively 1', 2, 3', 4', &c. the sines of 2', 3', 4', 5', &c. in their order will be calculated.

The numbers thus obtained will at first scarcely differ from an uniform progression, the versed sine of 1', which forms the multiplier of deviation, being so extremely small. It is hence superfluous, to compute rigidly all those minute variations. The labour may be greatly shortened, by calculating the sines for each degree only, and employing some abridged process for filling up the sines, corresponding to the subdivision in minutes,

The arc of one degree being equal to .0174533, it follows from the scholium to Prop. 6., that the sine of 1°=.0174533—(.0174533)3 =.0174524, and hence the versed sine of 1°=(.0174524)=.0001523. Wherefore sin(A+1o)=2sinA-2sinA x .0001523—sin(A-1o); or, if from twice the sine of an arc, diminished by its 6566th part, the sine of an arc one degree lower be subtracted, the remainder will exhibit the sine of an arc, which is one degree higher. Thus,

Sin2°-2sin1-2sin1° x .0001523.0349048-.0000053 =.0348995.

Sin3°-2sin2° -2sin2° X.0001523- sin1°.0697990

.0000106-.0174524.0523360.

Sin42sin3°—2sin3o × .0001523—sin2°.1046720.0000160.0348995.0697565.

After this manner, the sine for each degree is computed

in succession.

But the sines may be found, independently of the previous quadrature of the circle. Assuming an arc whose chord is already known, it is easy, from Prop. 6. to determine the successive chords and supplemental chords of its continued bisection. Let that arc be 60°; its chord is equal to the radius, and (IV. 17. cor. 2.) its suppleinental chord 31.7320508076. Whence the supplemental chord of 30°=√(2+1.7320508076)=1.9318516525. In this way, by continued extractions, the supplemental chords of 15o, 7° 30′, 3° 45', and 1° 52' are successively computed, the last one being equal to 1.9997322758. Again, the chords themselves are deduced by a series of analogies; for 1.9318516525: 1:1.51763809004-chord of 30°, and so repeatedly, till the chord of 1° 52', which is .0327234633. Hence, taking the halves of those numbers, the sine of 56'4.0163617317 and the cosine of 56'4 9998661379, and therefore (cor. 3. defin.) the tangent of that arc is .0163639215; consequently the arc itself

(2x.0163617317+.0163639215)=.0163624616, and thence the length of the arc of a minute is .0002908882086. Wherefore the sine of 1'=.0002908882.0002908882)3 =.00029088826046, and the versed sine of 1'=

(.00029088826046)*.000000042308.

Employing these data, therefore,

Sin2' 2sin l'-2sin l'x.000000042308.0005817763845; Sin3'2sin2'-2sin2' x .000000042308—sin1'=

.0008726645152; and so forth.

But it is very seldom requisite to push the estimation to such extreme nicety. The sines being calculated for each degree as before, those corresponding to the subdivi

120

sion in minutes, may be found by a more expeditious method, though founded on ulterior considerations. If the sines increased uniformly, the sine of A°+n' would exceed that of A by the quantity(sinA+1°-sinA—1°)=B. But the rate of this augmentation, being continually retarded, occasions a defect, equalton2 × sinA × .000,000,042308=C. Again, since the retardation itself gradually relaxes, it requires a small compensation, which may be estimated at (60-n')B x.0000013 D. The sine of A°+n' is then very nearly sinA+B-C+D. Thus, the sines of 310, 32°, and 33° being respectively .5150381, .5299193, and .5446390, let it be required to find the sine of 32° 40'.

40

Here B ;(sin33° — sin31o) = .0098670,

120

= = .0000003.

C= 1600 × sin32 × .0000000423.0000359, and D=20 x .0098670 × .0000013 Whence sin32° 40'.5299193+.0098670.0000359+ .0000003 = .5397507.

After the sines are calculated up to 60°, the rest are deduced from cor: 4. Prop. 3. by simple addition. Thus, sin61° sin59°+sin1°.8571673+.0174524.8746197.

The VERSED SINES and supplementary versed sines are only the difference and sum of the radius and the sines.

The TANGENTS are easily derived from the sines, by help of the analogy given in the third corollary to the definitions. Thus, cos32°: sin32:: R: tan32o, or, .8480481 :.5299193 ::1:.6248694=tan32°. Beyond 45°, the calculation is simplified, the radius being (cor. 7. defin.) a mean proportional between the tangent and cotangent, or the cotangent is the reciprocal of the tangent.

The SECANTS are deduced by cor. 4. to the definitions, since they are the reciprocals of the cosines.

From the lower tangents and secants, the tangents of arcs that exceed 45° are most easily derived; for (cor. 4. Prop. 4.) tan(45°+a) sec 2a+tan 2a. Thus, tan46° sec2°+tan2°, or 1.0355303=1.0006095+.0349208.

=

PROP. VII. THEOR.

In a right angled triangle, the radius is to the sine of an oblique angle, as the hypotenuse to the opposite side.

Let the triangle ABC be right angled at B; then R: sinCAB:: AC: CB.

For assume AR equal to the given radius, describe the arc RD, and draw the perpendicu

lar RS. The triangles ARS and ACB are evidently similar, and therefore AR RS :: AC: CB. But, AR being the radius, RS is the sine of the arc RD which measures

A

S

B

the angle RAD or CAB; and consequently R: sinA :: AC: CB.

Cor. Hence the radius is to the cosine of an angle, as the hypotenuse to the adjacent side; for R: sinC or cosA :: AC: AB.