SC I E N C E -Supplement. 



FRIDAY, JANUARY 1, 1886. 



THE STUDY OF GEOMETRY. 



We have a pernicious habit in this country of 

 supposing, that, because in a republic all men are 

 born equal as to their rights, they are also born 

 equal as to then abilities. We have a different 

 theory in regard to horses : we know that a race- 

 horse is altogether different from a dray-horse, 

 and we give him a totally different kind of life 

 from the beginning. We have no trouble in recog- 

 nizing him : we simply inquire who were his an- 

 cestors, and our expectation as to his qualities is 

 carefully based upon the answer to that 'question. 

 It would, perhaps, be a good plan if the young of 

 the human species were divided into two groups at 

 an early age, — one large and one small ; one com- 

 posed of those of whom nothing more than plain 

 living is expected, and the other composed of the 

 race-horses, of those whose ancestors, or whose 

 chance endowments, give reason to hope that they 

 may give some aid to learning or to culture. 



There is, at all events, no reason why all young 

 people should be taught geometry in the same 

 way. For most children, a form of reasoning so 

 abstract is not only repulsive, but very nearly im- 

 possible of comprehension. A little may be done 

 for them (or for then descendants) by giving them 

 a small dose of geometry, made as plain and easT 

 and direct as it can be made; but they do not need 

 to know every thing that can be done with the 

 straight line and circle. Life is short, and the 

 whole content of geometry as known to Euclid 

 is long. For most children in schools, a good 

 specimen of the kind of reasoning, and a fair 

 knowledge of the principal results, are all that is 

 desirable. For such, a geometry like Wentworth's 

 serves a very good purpose. 



But it is a pity that the kind of geometry a per- 

 son is taught should depend upon his geographical 

 position near this or that kind of a school. Any 

 one whose destiny is to do difficult thinking in 

 after-life should have a different kind of early 

 training: he should dwell long among the geo- 

 metrical concepts, should become thoroughly im- 

 bued with the bare and rigid form of reasoning, 

 and should have the results as familiar as his 

 mother-tongue. It is a serious loss to him if he is 

 made to run over the subject with uncouth haste. 

 Students of this kind will find their natural guide 



in such a text-book as Newcomb's or Halsted's. 1 

 In neither is it the aim to give the most rapid and 

 cursory system possible. Both are written from 

 the stand-point of the modern idea that the geome- 

 try of this world is not the only possible geometry, 

 and that it is mere matter of accident that two 

 parallel lines do not approach each other, and that 

 two straight lines do not enclose a space. Both 

 have felt the influence of the syllabus of the 

 English association for the improvement of geo- 

 metrical teaching. The idea of figure is shorn of 

 its material content, and limited to its bounding 

 lines or surfaces. The sum of two right angles is 

 not regarded as a purely imaginary idea with no 

 reality corresponding to it, but the ' straight angle ' 

 is allowed to play its natural part. In Professor 

 Newcomb's book, nis favorite idea is carried out 

 of leading up to new and strange conceptions by 

 very slow and gradual steps : Mr. Halsted's is in- 

 tended for boys 2 of much more highly developed 

 minds. There are no concessions to youthful 

 weakness. It is also intended for boys of well- 

 developed taste in the art of book-making. It 

 presents a splendor of paper and of margin which 

 is far removed from the republican simplicity of 

 our ancestors. 



The ancients believed that the geometrical con- 

 cepts came down from heaven, but that the chief 

 end of geometry was to measure the earth. We 

 admit now that the concepts are, in the first 

 instance, of the earth and earthy; but we have 

 given an enormous development to the geometry 

 of pure position, and have made it as remote from 

 all possibility of application as the theory of num- 

 bers itself. It is in consonance with this develop- 

 ment that in both these books measurement is 

 given somewhat the position of an appendix to the 

 subject, instead of being made to appear as the end 

 towards which all the propositions lead up. 



Mr. Halsted does an excellent thing in giving an 

 introductory chapter on logic. When pure reason- 

 ing is about to become the student's daily occupa- 

 tion for many months, it is a pity not to give him a 

 general view of the processes involved at the start. It 



1 The elements of geometry. By George Bruce Hal- 

 sted. New York, Wiley, 1885. 8°. 



2 As a synonyme for ' student of geometry,' one should, 

 however, say girl with the understanding that boys are to 

 be included. Geometry is chiefly studied in the high 

 schools, and the high-school graduates number three girls 

 to every boy. If geometry is as good a specific against bad 

 reasoning as is commonly supposed, logicalness will soon 

 become a feminine instead of a masculine characteristic. 



