452 Inxh-ifun of tht BliJer on GuiiUr't RuU. 



TJic next opf r.ition which offers itfclf here is reciprocal proportion, which can be effcdled 

 by r.D other niethoil than by inverting the Aider, but which is rendered as eafy by this ap- 

 plication, as dirert proportion is in the common way ; for if any antecedent number on 

 B inverted be fet to its confcquent on A, any other antecedent on B, in the fame pofilion, 

 will ftand againft its confequcnt on A, fo as that the terms may be in a reciprocal ratio. 

 In fi)uarliig any number, it will appear from what has been already f;iid, that if the number 

 to be fquared be placed on B, inverted ngainfltlie fame on A, the fquare will ftand on B, 

 againfl unity on A. Therefore, to extraft the fquare root of any number, let that number 

 on B fl^nd againft unity on A ; and then wherever the coincident numbers are both of the 

 fame value, that point indicates the root. If two dividing lines of the fame value do not 

 CTtaflly coincide, the coincident point will be at the middle of the fpace contained between 

 thofe two which are neareil a coincidence ; and as there is only one fuch point, there can be 

 no miilake in readily afccrtaining it. The finding of a mean proportional between any 

 two numbers is extremely eafy at one operation ; for if one of the numbers on B inverted 

 be fet to the other on A, the coincident point of two fimilar numbers (hews either of thofe 

 to be the mean, or fquare root of their product, according to the preceding proccfs. Thus 

 have we a (hort and eafy method of multiplying, dividing, working reciprocal proportion, 

 fquaring and extracting the fquare root, at one pofition of the inverted Aider, whereby 

 the eye is directed to only one point of view for the refult, after the Aider is fixed: where- 

 as, by the common method of extracting the fquare root by A and B dire£l, the Aider re- 

 quires to be moved backwards and forwards by adjullmer.t, the eye moving alternately to 

 two points, till fimilar numbers ftand, one on B againft unity on A, and the other on A 

 againft the fquare number on B ; which fquare number, in the cafe of finding a mean pro- 

 portional, mull be found by a previous operation. Hence, for more convenience in the 

 extraction of roots, and meafuring of folids, an additional line called D, has been added 

 to the rule, which renders it more complex, and confequently feldom underftood by an 

 artificer. Upon examining the patent rule with a graduated piece of brafs, to be attached 

 to, or detached from the Aide at pleafure ; and alfo Mr. Horton's new improved rule with 

 two Aiders; I perceive that the method of multiplying length, breadth, and depth into one 

 another at one pofition, in which their excellence confifts, is nothing more than a com- 

 bination of the two methods on the common rule ; for the producft of the two firft 

 numbers, effected by means of an inverted and a direft line, forms the multiplier to be 

 ufed with the third number, by a third direft line, at the fame pofition of the Aiders. 

 Hence, a ganger's rule will anfwer the fame purpofe, if an additional Aider be held in a 

 proper fituation inverted. No reafon, however, has been given for the effe£l of this procefs 

 in any book that I have yet feen, nor of the application of an inverted Aider to any other 

 fpecies of calculation *. 



• In reply to tlic obliging poftfcript of the author of this paper, I have only to remark, that the appro- 

 bation of the intelligent cultivators of fcience is the moll ellimahic reward ef my endeavours to diffufc 

 knowledge ; and ihat inclination, as well as duty, will induce me v> pay every attention to his future com- 

 munications. N. 



V. Abjlraa 



