442 THE POPULAR SCIENCE MONTHLY. 



But when we examine his argument we find that he has made three 

 unproved assumptions namely : 1. That a thing cannot at the same 

 time he and not he j 2. That if equals be added to equals, the wholes 

 are equal ; 3. That things which are equal to the same are equal to 

 one another. It so happens that each of these propositions which he 

 has assumed to be true is, if true, much more important than the 

 proposition which he has joroved. Let us point out these three as- 

 sumptions to our bright student, and then resume our catechism. 



Q. Could you possibly prove this pro2)osition in geometry if any 

 one of those three assumed propositions were not granted ? 



A. No. 



Q. Then, if we deny these assumptions, can you prove them ? 



A. No ; but can you deny them ? 



No, we cannot deny them, and cannot prove them ; but we be- 

 lieve them, and therefore have granted them to you for argument, 

 and know your projDOsition of the two right angles to be true, because 

 you have proved it. 



Now, here is the proposition which Euclid selected as the simplest 

 of all demonstrable theorems of geometry, in the demonstration of 

 which the logical understanding of a student cannot take the first step 

 without the aid of faith. 



From the student let us go to the master. We go to such a teacher 

 as Euclid, and in the beginning he requires us to believe three propo- 

 sitions, without which there can be no geometry, but which have 

 never been proved, and, in the nature of things, it would seem never 

 could be proved namely, that space is infinite in extent, that space 

 is infinitely divisible, and that space is infinitely continuous. And 

 we believe them, and use that faith as knowledge, and no more dis- 

 trust it than we do the results of our logical understandings, and are 

 obliged to admit that geometry lays its broad foundations on our 

 faith. 



Now, geometry is the science which treats of forms in their rela- 

 tions in space. The value of such a science for intellectual culture 

 and practical life must be indescribably important, as might be shown 

 in a million of instances. No form can exist without boundaries, no 

 boundaries without lines, no line without points. The beginning of 

 geometric knowledge, then, lies in knowing what a " point " is, the 

 existence of forms depending, it is said, upon the motion of points. 

 The first utterance of geometry, therefore, must be a definition of a 

 point. And here it is : " A point is that which has no parts, or which 

 has no magnitude." At the threshold of this science we meet with a 

 mystery. "A point is" then, it has existence "is" what? In 

 fact, in form, in substance, it is nothing. A logical definition requires 

 that the genits and differentia shall be given. "What is the genus of a 

 " point ? " Position, of course. Its differentia is plainly seen. It is 

 distinguished from every thing else in this, that every thing else is 



