of Weights and Meamres. 



243 



24-5 inches, i 

 H. diam. 



Z-^>^J 42 inches 

 length. 



3r5 inches, B. diam. 



of which the hypoth. or diag. eh will be 35 inches, and the 

 perp. e d will be 28 inches, as we will prove thus : 



a b and c d are respectively equal to one another ; and e c 

 and df are also equal to one another : therefore, if re be added 

 tx> ab, the sum will be ed. Or, ec is half the difference be- 

 tween ab and ^^ consequently ed is the arithmetical mean 



between 31-5 and 24"5, and -^^-^~ — = 28 = ed; [and since 



21 and 28 are the two sides of a right angled triangle, includ- 

 hig the right angle, the diagonal cb will, according to the 4'7th 



prqo. 1st Euclid, be ^^^4-21'*= 35; for the sum of the 

 squares of 28 and 21 make 1225, to extract which by logarithms 

 we have 



1225, its log. =2) 3-0881361 

 its root 35, its log. = 1 •54-4'0680 ; therefore, 

 100*78 ale gallons ought to be found upon the rod, among the 

 engravuigs, at the 35th inch; instead of which, if we followr. 

 Overley, we have 



galls, diag. galls. diag. 



As 60 : 30^': : 100-78 : .y45351 = 35-66 instead of 35; for 

 by logarithms 



^453577 its log. = 3)4--6 565869 

 its root 35-66^ its log. = 1-5521956; and from this tmalogy 

 the diagonal rods, now in use, are constructed. We will now 

 see what the diagonal ol' this 60 gallon cask ought to have 

 been uistcad of 30 inclies, according to Overley: tlius, 

 galls. diag. galls, inches. 



As 100-78 : 35^': : 60 : 4^25525~89899T= 20'i5. 

 For 2.5525-«f)8995, its log. S)i-4()6f)810 



its root 29-4.5, its l"g- = ]-i(iW)9:i7, which is different from 



H h 2 thftt 



