84:8 Dynamic Theory. 



a postulate as a problem whose solution is self-evident. A theorem in 

 geometry " is a truth requiring demonstration." Axioms and postulates 

 are things which are perceived as soon as seen. For example, it is an 

 axiom that the whole is greater than a part ; another, that a straight 

 line is the shortest distance between two points ; again, that things 

 which are equal to the same thing are equal to each other, &c. These 

 propositions are seen to be so, and the seeing is the end of the demon- 

 stration. Nothing can go beyond it. The demonstration which a the- 

 orem requires, is such a picking to pieces, or analysis, that the parts 

 can be seen. It is a reduction of the theorem to axioms. For example, 

 let it be demonstrated that in any triangle the sum of the three angles is 

 equal to two right angles. Let X, fig. 380, represent any triangle ; 

 then the angles a . -j- b -j- c equal two right angles, or 180 degrees. Pro- 

 duce the sides C and D so as to form the external angle d opposite 6, 

 and draw E through 0, parallel with the side C , and produce the side A 

 to B. All the angles which it is possible to construct about a given 

 point, as 0, must together equal the circle, or 360, or four right angles. 



and any straight line, as A B, 

 passing through 0, must split 

 the circle exactly into equal 

 parts, so that the sum of the 

 angles about such point on one 

 side of such straight line will 

 equal 180, or two right angles, 

 therefore c -|- e -)-/= two right 

 angles. The lines C and E be- 

 FIG- 380. ing parallel, the} 7 must meet A 



B at the same angle, so that the angle a = the angle / ; C and E also 

 meet D at the same angle, consequently the angle d = e. But d is equal 

 to its opposite angle b, so that b = e. Consequently a, b and c being 

 equal to /, e and c, they are equal to two right angles ; which was to be 

 proved. Thus, this theorem, not obvious at first, as a whole, is found 

 to involve a number of propositions, each of which is 

 obvious to most perceptions. But if one should hesi- 

 tate to admit some of the assumptions, as, for exam- 

 ple, that the opposite angles, d and 6, are equal, that 

 proposition becomes a theorem, and may be further re- 

 duced. Let a figure ( 381 ) be constructed by drawing 

 two straight lines across each other so as to form op- 

 posite angles. Then will the sum of the angles g and 

 d equal two right angles; g plus b will likewise equal two right angles; 

 g -j- d therefore equals g -\- b. Now, if g be taken from each of these 

 equal quantities, the remainders will be equal ; that is, the angle d will 



FIG. 381. 



