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PROJECT FOR A NEW

TRACT 29.

18. From the sines and cosines also, the tangents, co-tangents, secants, and cosecants, may he made by theseknown proportions, viz, as

1. cosine

2. sine

3. cosine

4. sine

5. radius

6. radius

7. tangent

radius :

: sine :

tangent,

radius :

: cosine

cotangent

radius :

: radius :

secant,

radius :

: radius :

cosecant,

sine :

: secant :

tangent,

cosine :

: cosecant:

cotangent.

radius :

: radius :

cotangent.

Therefore, the reciprocal of the cosine will be the secant;the reciprocal of the sine, the cosecant; the quotient of thesine by the cosine, the tangent; and the quotient of thecosine by -the sine, the cotangent; or the product of thesine and secant will be the tangent, and the product of thecosine and cosecant, the cotangent; or, lastly, the recipro-cal of the tangent is the cotangent; proper regard beinghad to the number of decimals, on account of our radiusbeing 100000, instead of 1 only.

And these are to be used when the application happens tobe easier than the general series, and easier than by propor-tion from Rheticus’s canon.

But there are other particular theorems, which, by a littleaddress, may be rendered more expeditious titan any of theformer: thus,

19. In any two arcs, this is a general proportion ;

As the difference of their sines:to the stun of their sines ::so tangent of half the difference of the arcs :to tangent of half their sum.

So that, by taking continually the arcs, having the commondifference 2, the third term of this proportion will be 1,and the fourth term will be found by dividing the sum of thesines by their difference, which divisor, or difference, will