Add the second and fourth together, and substitute B for A,

and A for B as before: then,

cos B+A+cos B-A≈2 cos 4. cos B; that is,

cos B+ A=2 cos A. cos B-cos B—A (Z)

Let n—1.A=B; this value being substituted for B in the expressions Y and Z, we have the two following theorems for the sines and co-sines of multiple arcs, viz.

Theor. 1. Sin nA=2 cos A. sin n- 1 A-sin n-2A.

2. Cos n=2 cos A. cos n-1 A-cos n-2 A.

In which general theorems, if n be expounded by 1, 2, 3, 4, 5, &c. we have the formulæ for all particular multiple arcs, viz. if

n=2.

5. Sin 24-2 cos A. sin A (from theor. 1.)

6. Cos 24=2 cos A. cos A-cos 0 (= 1) (theor. 2.)

.{

7. Sin 34=2 cos A. sin 24-sin A (theor. 1.)

n=3.

8. Cos 34-2 cos A. cos 24-cos A (theor. 2.)

n=4.

n=5.

&c.

{

{

9. Sin 44 2 cos A. sin 34-sin 2A (theor. 1.) 10. Cos 44 2 cos A. cos 34-cos 24 (theor. 2.) 11. Sin 542 cos A. sin 44-sin 34 (theor. 1.) 12. Cos 5A=2 cos A. cos 44-cos 34 (theor. 2.)

46. These formulæ may be continued to any length, and by means of them, the sine and co-sine of every degree and minute of the quadrant, may be computed, as will be shewn; but, hav.. ing found the sines and co-sines to the end of the first 30 degrees by this method, those from 30° to 60° may be obtained by an easier process, by means of the following formula.

Add formula 1 and 3 (Art. 44.) together, and sine 4+ B +sin A—B=2 sin A. cos B; let A=30°, then will sin. A=4 (cor. Art. 34); substitute these values of A and sin. A in the above expression, and it will become

sin 300+B+ sin 30-B (2 x x cos B) cos B;

Formula 13. sin 30+ B=cos B-sin 30-B,

47. The tangents of two unequal arcs A and B being given, to find the tangents and co-tangents of their sum and difference, It has been shewn (Art. 35.), that when radius=1, the

tangent of any are=

sine

co-sine

wherefore, by substituting for

the sine and co-sine their respective values as given in the formulæ, Art. 44. we shall have

If both terms of the right hand fractions be divided by

cos A. cos B, they will become

48. To find the tangents and co-tangents of multiple arcs; that is, if A be any arc, to find the tangent and co-tangent of nA.

49. These formulæ may be extended to every minute of the quadrant; but although it seemed necessary to shew how the tangents and co-tangents of multiple arcs are expressed in terms of the tangents of the component arcs themselves, yet we have shewn how to compute the tangents and co-tangents for the first 45° by means of the sines and co-sines, which is in many respects preferable to the above method. The tangents and co-tangents of arcs above 45°, may be found by a very easy process, the formula for which is deduced as follows:

It appears

from formulæ 16 and 17, Art. 47. that
tan A+tan B

; let A=45°, then (Art. 16. cor.)

Tan A+B tan. 4=1.

1tan 4. tan B

for this fraction substitute its equal (2 tan 2B) in the last equation but one, and we shall have tan 45°+ B-tan 45°-B= 2 fan. 2B; hence arises

Formula 26. Tan 45°+B=tan 45°-B+2 tan 2B ".

THE METHOD OF CONSTRUCTING A TABLE OF SINES, TANGENTS, SECANTS, AND VERSED SINES.

50. In the preceding articles the methods of deriving expressions for the sines, co-sines, tangents, &c. of the sum, difference, and multiples of arcs in terms of the sines, co-sines, &c. of the arcs themselves, have been shewn; but before we can employ these formulæ in the actual construction of the trigonometrical canon, in which the numerical values of the sine, tangent, &c. of arcs for every minute of the quadrant are usually exhibited, it will be necessary to compute the sine and co-sine of 1 minute, and from these we shall be able, by means of what has already been proved, to determine not only the numerical values of the rest of the sines and co-sines, but likewise those of the tangents, co-tangents, secants, co-secants, versed sines, and co-versed sines, which constitute the entire canon.

51, To find the sine and co-sine of an arc of 1', the radius being unity.

It has been shewn (part 8. p. 231, 232.) that if the radius of a circle be unity, the semi-circumference will be 3.1415926535898 nearly; this semi-circumference consists of 180 degrees, each degree being 60 minutes; that is, of (180 × 60=) 3.1415926535898 10800 minutes; 10800

=.0002908882086=the

length of an are of 1', the radius being unity.

But in a very small arc, as that of 1', the sine coincides indefinitely near with the arc "; wherefore the above number

m The trigonometrical formulæ, introduced into this work, are those only which are necessary for the construction of a table of sines, tangents, &c. Several of the French and German mathematicians have excelled in this species of investigation, and produced a great variety of theorems suited to every case in Trigonometry. The English reader will find a collection of formulæ, applicable to the most delicate investigations in Mechanics, Astronomy, &c. in Mr. Bonnycastle's Treatise on Plane and Spherical Trigonometry, London, 1806.

"In Simpson's Doctrine and Application of Fluxions, part 2. p. 501, and

.0002908882, &c. may be taken for the length of the sine of l'. Wherefore also (Art. 35.) the co-sine of 1'=√1-sin_1')2= (✓.99999991538405, &c.=) .99999996.

52. Construction of the sines and co-sines from 0 to 30o.

Since (Art. 51.) the sine of 1'=(.0002908882086, &c.=) .0002909, which is its nearest value to seven places of decimals, and co-sine of 1'=.99999996. Let A=an are of 1', then the above numeral values being substituted respectively for sine and co-sine of l' in formula 5. Art. 45. we shall have

By Formula 5. sin 2' 2 cos 1'. sin l'=2x .99999996 × .0002909=.0005813, here the sine of 2' is found

F. 6. Cos 2'-2 cos 1'. cos 1'-1=2x .99999996 x .99999996 —1=.9999998, here the co-sine of 2′ is found.

F. 7. Sin 3′=2 cos 1'. sin 2'-sin l'= 2 x .99999996 × .0005818-.0002909=.0008727, here the sine of 3′ is found.

F. 8. Cos 3'=2 cos 1'. cos 2'-cos 1' 2 x .99999996 X .9999998—.99999996=.9999996, here the co-sine of 3′ is found.

F. 9. Sin 42 cos 1'. sin 3'-sin 2=2 × .99999996 × .0008727-.0005818=.0011636.

F. 10. Cos 4'=2 cos 1'. cos 3'-cos 2'2 x .99999996 X .9999996-.9999998.9999993.

F. 11. Sin 5' 2 cos l'. sin 4'-sin 3′.0014544.

F. 12. Cos 5'=2 cos 1'. cos 4'-cos 3'=.9999989. And in this manner proceed to find the sine and co-sine of every minute as far as 30°.

52. B. To find the sines and co-sines from 30° to 60o.

By formula 13. Art. 46. sin 30°+B=cos B-sin 30-B.

in Vince's Fluxions, p. 220. it is shewn that (radius being 1,) the sine of any A3 A5 A7 2.3 2.3.4.5 2.3.4.5.6.7

arc A=A-: +

+ &c. = (in the present instance)

+ &c. =.0002908881676, &c. the sine of 1', which differs from the above expression for the length of the arc of 1' by only .000000000141; that is, the arc of l' and its sine, coincide to 9 decimal places inclusive, therefore the sine of 1' to 7 places of decimals (the number to which the tables are usually computed) exactly coincides with its arc.