To the Editor of the Popular Science Monthly.
I AM a great admirer of Herbert Spencer, and especially of his wonderful "Answers to Criticisms" in your journal. When he seems entirely caught and inwoven by his adversaries, with one blow of his trenchant blade he cuts the net, and is free.
He is one of the highest of living authorities, and I read with deep attention his two editions of "The Classification of the Sciences," being particularly interested in Table I., "The Abstract Sciences." All of it but two divisions he devotes to mathematics as exactly equivalent to quantitative relations; still, at the present day, it seems an untenable cramping of mathematics to define it as the science of quantity.
A candid note in Mr. Spencer's first edition shows that it was not till after he had actually drawn up this table that he became aware of one of the most important points in the question to be solved.
It is a note to his first great division of mathematics, and says: "I was ignorant of the existence of this as a separate division of mathematics, until it was described to me by Mr. Hirst, whom I have also to thank for pointing out the omission of the subdivision 'Kinematics.' It was only when seeking to affiliate and define 'Descriptive Geometry' that I reached the conclusion that there is a negatively-quantitative mathematics as well as a positively-quantitative mathematics."
All this confession is omitted in the second edition, where, however, the much superior expression "Geometry of Position" is substituted in the table for "Descriptive Geometry," which latter was very apt to be misleading, especially to engineers, from its technical sense, in which sense, of course, Spencer did not mean it.
Now let us try to explain, in few words, what the problem was that Hirst so unexpectedly put before Spencer's mind, that you may judge whether "seeking to affiliate" it to a scheme already drawn up was a proper mental condition in which to deal with a question so important, so subtile, so profound.
Geometry, as the abstract science of space, naturally resolves itself into two great divisions, geometry of measurement and geometry of position—geometry quantitative or metrical, and geometry morphological or positional.
As an example of the first, we may take the most ordinary illustration, that of equivalent triangles. Any two triangles having the same base, and their vertices in a line parallel to that base, will be of equal or "equivalent" superficial magnitude. Although the sum of the three sides of the one triangle might be a thousand times as great as the sum of the three sides of the other, they will contain the same number of square inches or square feet. This is a metrical or quantitative proposition; but, on the other hand, many propositions are known which are purely descriptive or morphological. Take the one, perhaps, best known, the celebrated hexagram.
In any circle join any six points of the circumference by consecutive straight lines in any order: the intersections of the three pairs of opposite sides are in a straight line. Or, take any two straight lines in a plane, and draw at random other straight lines traversing in a zigzag fashion between them, so as to obtain a twisted hexagon or sort of cat's-cradle figure: if you consider the six lines so drawn symmetrically in couples, then, no matter how the points have been selected on the given lines, the three points through which these three couples of lines respectively pass will lie all in one and the same straight line. So great an authority as Prof. Sylvester has stated that this proposition "refers solely to position, and neither invokes nor involves the idea of quantity or magnitude." Take another: If any pencil of four rays is cut by a transversal, any anharmonic ratio of the four points of intersection is constant for all positions of the transversal.
Now, Carnot in his splendid "Geometry of Position," and many before and after him, have laid open a whole world of truths of this kind, truths undeniably geometrical in their nature, but founded on the primitive idea of position, and bringing in any idea of quantity only incidentally and afterward. Now, this was evidently a branch of mathematics, but, having made his scheme mathematics only coextensive with quantitative relations, Herbert Spencer must force this under the quantitative rubric, and thus was betrayed into error. Seeing that it was not really positively quantitative, he could only call it negatively quantitative, but in doing this entirely misrepresents it. In Table I. he has, under "Abstract Science:"