^x o Geometrical InflruBions 



AXIOM V. 



// from Things unequal one tales Tkinis equal^ the Remain- 

 der P^dl he meqaau 

 If from the unequal Lines A E, A E, 



One take away the equal Line A D, A D, 

 The remaining Part D E, D E, 



willbeunequaL 

 Exphnation. This is the very Reverfe of the 4th Axi- 

 om; lince, if the equal Number, or Line of two Foot, be 

 taken.from the whole unequal Line, or Number of feven 

 Foot, 'tis certain, the unequal Number of five will remain, 

 "which is the Purport of this Axiom. 



AXIOM VI. 



The Things that are donhle to one another^ are equal among^ 

 them/elves. 

 Thus the Lines D D, D D, 



Which are double to the Line AD, ; 



are equal between themfelves. 

 Explanation, This is demonftrated by the Lines in Fig. 

 y, where the Lines D D. of 40 Foot long, tho* they are 

 double the Li"e D A, are iieverthelefs equal amongft 

 themfelves, 



AXIOM VJL 



Jhe Things vohich are the half of one and the Jame, or of 



Things equal are uneqnalanwngji themfelves. 



Thus the Lines A D, A D, 



Which are the half of the Lines D D, D D, 

 are equal between tlicmfelves. 

 Explanation, This is again the Reverfe of the lafi Axi- 

 om y {viz.. the Vlth) for tho* the Lines D D ; are double 

 to the Lines A D, A D, yet they, are neverthelefs equal 

 amongft themfelves. 



That vphich is faid of Lines may he alfo [aid of Superficies 

 And Solids j hovpever trivial thefe Things may appear, 'tis on 

 thefe our Mahematical Dif put ants ground their y^rguments ; 

 and tho they arc not very much us'd in our Way, they could 

 not poffillj he p^Js'd over wit horn manifeft Injury to this Suh- 

 jett. 



