VOL. XLVm.] PHILOSOPHICAL TRANSACTIONS. 339 



And this is the reason, why the sine of 90°, and the tangent of 45°, are limited 

 by the same termination as the 2d radius on the line of numbers. 

 To construct the line of logarithmic sines. 



From the scale of equal parts, take the numbers expressing the arithmetical 

 complements of the log-sines of the successive degrees, and parts of degrees, in- 

 tended to be put on the scale, descending orderly from 90° : then these distances 

 successively laid from the mark representing 90° at the right-hand end of the 

 scale, will give the several divisions of a scale of logarithmic sines. For, the ends 

 of any scale being assigned, the progressive divisions of that scale are laid on it 

 from that end which represents the beginning of the progression : or, the same 

 divisions may be laid from the other end, by taking the complements of the 

 terms to the whole length of the scale : consequently the arithmetical com- 

 plements of the sines are to be laid from the division representing 90 

 degrees. 



To construct the line of logarithmic tangents. 



These are laid down in the same manner, and for the same reasons, that the 

 sines were ; the tangent of 45° standing against the sine of 90°. The divisions 

 for the tangents above 45°, are reckoned on the same line from 45° towards the 

 left hand ; or any tangent arid its co-tangent are expressed by the same division. 

 Thus one mark serves for 40° and 50° ; and the division at 30° serves also for 

 60°; that at 20° serves for 70°, &c. and the like is to be understood of the in- 

 termediate divisions. For, as the tangent of an arc, is to radius ; so is radius, 

 to the co-tangent of that arc. Therefore the tangent is equal to the square of 

 radius divided by the co-tangent. And the co-tangent is equal to the square of 

 radius divided by the tangent. 



Now the radius being unity, its square is also unity. Therefore the tangent 

 and co-tangent of any arc are the reciprocals one of the other. But the recipro- 

 cals of numbers are correlatives to the arithmetical complements of their loga- 

 rithms. Therefore the logarithms of a tangent and its co-tangent, are arithmetical 

 complements one of the other ; and consequently will fall at equal distances 

 from 45 degrees. Therefore, in the line of logarithmic tangents, the divisions 

 to degrees under 45, serve also for those above ; both being equally distant from 

 45 degrees. 



To construct the line of logarithmic versed sines. 



As the greatest number of degrees will fall within the limits of the scale, by 

 beginning at 180°; therefore the termination of this line is at 180°, which is put 

 against 90° on the sines : and though the numbers annexed to the divisions in- 

 crease in the order from right to left, yet they are only the supplements of the 

 versed sines themselves. Now subtract the logarithmic versed sines, of such 



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