tract 29.

DIVISION OP THE QUADRANT.

1.29

with all its decimal places, by the continual addition of2'06264807, the old tables may be converted into the new,by allowing for the odd seconds and decimals. And for thispurpose it will, perhaps, be best to use the large table ofRheticus , which contains the sines, tangents, and seconds,to ten places of figures for every 10", and also the differ-ences. At least, such sines, &c, may be found in this way,as have their seconds and decimals well adapted for thepurpose ; and for such as would be found too troublesomein this way, recourse may be had to some of the followingmethods.

12. Let us now examine the expressions for the sines,8tc, by infinite series.

The radius being 1, and arc a , it is well known that thesine is = a -\a? + a> + a* - &c.

cosine =1 + ~ a* - ~ a 6 + - 1 — « 8 - &c.

O 17 f .a

tang. .= « + T fl3 +— a* + a 7 + w + &c.

i

45

720

17

315

2

945

2835

1

4725

277

1 2

cotang. si ar' — la —— a 3 - : — a 5

° 45 94o

secant =1 +.^+ 1 f-a 4 +a 6 + 8Q64

7 31 1°7

cosec. = fl- 1 + {-« + a3 +i5igo aS+ «04800 ° 7 + &c *

a 7 — &.c.a 8 + &c.

Or the same series are thus otherwise expressed:

d

sine = a

l , , b .

-rrr 5 a

7 s -a 7 -|- a 9 — &c.

6.7 ‘8.9

d

^ nr ~i~ —- nr — -//- _i_ .

tangent = a + + See.

cosine = i - -i a 1 -)- -1- a 4 - — a 6 + ~ a* - &c.

3.4 5.o '7.8

cotang. = a 1 - fa - 1J a * ~ ff * s - ~ a 7

1385d

&C.

a4 +Co- S +3^ S + &C -

secant = I + 2 - ■ 13

«osec. = a -i + la + ^ a 8 + |5 a s + “« ,r + fcc.

you. II.

K