Reducing Practical Tables and Calculations. 



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If we ufe the fecond and third' proportions, putting two and three for-. 

 in and -n, and for a and b, three-halves and two; in the full cafe j .and-. 

 one-half and -one, in- the feeond we ^et the- fame values; ■ 



Hence in general 



r+ i r' r — i- r t 



x -fc — — N ;.N «.- x : : x : N- «x X. 



2 ' % ■ 



Another example of the advantage of transferring practices from one fub- 

 ject to another is this. L>r. Halley has applied a method fimilar to that" 

 of interpolation to mid the time of the tropicks : now the fun's meridian 

 alti Lide may be found in the fame way from altitudes taken near the meri- 

 dian, and -if the obferver begins a little before noon to take altitudes and the 

 times, and continues to do fo till a little afternoon, a number of meridian 

 altitude's may be deduced from thefe, and the latitude found much more ex- 

 actly from them, than can be expected from a fmgle meridian altitude, b^ 

 uung j the exprefhon forthe maximum, or othervvife; • 



Analogous to theTe, are methods of generalizing 

 properties from particular cafes : thus, if Ab Ac be" 

 tangents to a circle, and -if any lines BG, be, be alfo' 

 drawn to touch, thccircle j then^ the perimeters of all 

 the triangles ABC, will-be conftant, and alfo the dif- 

 ference between the fum of Ab and Ac and the bafe 

 be: this property is- of uncommon' ufe- in the coftftruc^ 

 tion of problems relative to plain triangles and trape- 

 ziums j and if lines be fuppofed drawn from the centre, m a point in the 

 circumference of a fphercy to each: part of the figure, it will be 

 found, that the projection of the figure upon the fphere will have analogo>is 

 properties, and that the theorem is alfo true in fpherical triangles. 

 By a like mode of confideration, problems fimilar to thofe of ■ Apcllo^i r s 



