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Excerpt from

THE COLLECTED MATHEMATICAL PAPERS OF JAMES JOSEPH SYLVESTER...
VOLUME II... Cambridge At the University Press 1908

James Joseph Sylvester,
A PROBATIONARY LECTURE ON GEOMETRY
DELIVERED...  4 DECEMBER, 1854.

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For obvious reasons also I think it would be inexpedient to attempt in this place a description of the difference between the spirit and methods of the ancient and those of the modern schools of geometry. I shall prefer to occupy the short remaining period of the lecture with inviting your attention to a distinction which lies deeper in the subject-matter of the science itself, being drawn from its relation to the two leading attributes of space, namely, magnitude and direction. I allude to a distinction, which or the like to which, runs through every branch of mathematical speculation, and has its analogue even in the study of the natural sciences, such as chemistry, botany and anatomy. When we have attained a certain elevation in our view of the subject, and can look down upon the territory which we have traversed to arrive there, we begin to perceive that geometry resolves itself naturally into two great divisions, geometry of position and geometry of measurement, geometry descriptive* or morphological and geometry quantitative or metrical. The ancients chiefly concerned themselves with the metrical properties of space; the more subtle and essential spirit of the science, however, probably resides in the purely descriptive part. A single proposition selected from each may serve to place the distinction between these two provinces of inquiry in a clearer light. If we draw any two triangles upon the same base, say for instance along this floor where the wall meets it, terminating respectively in two points, (so chosen that their line of junction shall be parallel to the base line) as for instance to two points in the line running along the cornice of the room, it is easily proved that the two triangles so formed, will be of equal superficial magnitude; this would be true although the apex of one of them were taken anywhere along the actual line of the ceiling, but the other in a prolongation of the cornice stretching out a hundred miles away. Both triangles so obtained would contain the same number of square inches or square feet, although the measure of one round its periphery might be a thousand times greater than that of the other. This is an example of a metrical or quantitative proposition. Again, if we take a triangle and bisect each side and join each bisecting point with the opposite angle, it is a known property of the triangle that these three lines must meet one another, not as three lines taken at hazard would do, cutting out another triangle between them, but in one and the same point. This proposition is partly metrical and partly descriptive; it is descriptive so far as regards the property of the bisecting lines passing through the same point; quantitative in so far as the idea of a line being bisected implies a notion of the relative magnitudes of the equal parts. Propositions however exist which are purely descriptive; as for instance, the celebrated theorem of Pascal known under the name of the Mystic Hexagram, which is, that if you take two straight lines in a plane, and draw at random other straight lines traversing in a zigzag fashion between them, from A in the first to B in the second, from B in the second to C in the first, from C in the first to D in the second, from D in the second to E in the first, from E in the first to F in the second and finally from F in the second back again to A the starting point in the first, so as to obtain ABCDEF a twisted hexagon, or sort of cat's-cradle figure and if you arrange the six lines so drawn symmetrically in three couples: viz. the 1st and 4th in one couple, the 2nd and 5th in a second couple, the 3rd

* The word " descriptive " is here employed out of its technical sense.

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and 6th in a third couple; then (no matter how the points ACE have been selected upon one of the given lines, and BDF upon the other) the three points through which these three couples of lines respectively pass, or to which they converge (as the case may be) will lie all in one and the same straight line. This is a purely descriptive proposition, it refers solely to position, and neither invokes nor involves the idea of magnitude. The existence, I will not say of a class, but of a whole world of truths of this kind, truths undeniably geometrical in their nature, serves to show how imperfect is the definition once generally accepted of geometry (however conformable to the etymology of the word and the early history of the subject), which described it as the science of the measurement of magnitude, in a word as a science of mensuration, which is in fact only one and that a subordinate division of the science. Sciences, true sciences, spring from celestial seeds sown in a mortal soil, they outgrow the restrictions which human shortsightedness seeks to impose upon them, and spread themselves outwards and upwards to the heavens from whence they derive their birth. We may write learnedly upon the history of geometry, upon its origin, growth, and apparent tendencies; but there is that within it which baffles our predictions and sets at nought our calculations as to the uses to which it may hereafter be turned and the form which it may be finally destined to assume; that which, analogous to the vital principle in an organized being, resists the circumscription of language and defies mere verbal definition. It has been said that to appreciate what virtue and morals mean, men must live virtuous and moral lives. It is equally true, that a knowledge of the objects of science is not to be attained by any scheme of definitions however carefully contrived. He who would know what geometry is, must venture boldly into its depths and learn to think and feel as a geometer. I believe that it is impossible to do this, to study geometry as it admits of being studied and am conscious it can be taught, without finding the reasoning invigorated, the invention quickened, the sentiment of the orderly and beautiful awakened and enhanced, and reverence for truth, the foundation of all integrity of character, converted into a fixed principle of the mental and moral constitution, according to the old and expressive adage "abeunt studia in mores."