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"The five chief solids of Plato, the Samian wise man invented,
And as Pythagoras found them, so Plato taught us their meaning:
This epigram from Psellus furnishes an indubitable confirmation of the views of Proclus and Kepler, respecting the arrangement and object of Euclid's great work.
I observed that, in former times, to study Euclid was to study geometry. This will serve as a sufficient apology for the space which I am bestowing upon the "Elements."
What was it, is the next inquiry, which caused the later mathematicians to vary so much from Euclid's course, and to omit whole books of his work? We will allow them to answer for themselves.
Of the first six books, and the eleventh and twelfth, Montücla remarks that they contain material which is universally necessary; and are to geometry what the alphabet is to reading and writing. The remaining books, he continues, have been considered less useful, since arithmetic has assumed a different shape, and since the theory of incommensurable magnitudes, and of the regular bodies, have had but few attractions for geometers. They are not however useless for persons with a real genius for mathematics. For these reasons, both Montücla and Lorenz recommend these five omitted books to mathematicians by profession. Of the tenth especially, Montücla says that it includes a theory of incommensurable bodies so profound that he doubts whether any geometer of our day would dare to follow Euclid through the obscure labyrinth. This observation is worth comparing with the expressions of Kepler and Ramus, above mentioned, on the same book.
Of the thirteenth book, which, with the two books of Hypsicles to follow it, treats of the regular solids, Montücla says, "Notwithstanding the small value of this book, an editor of Euclid, Foix,* Count de Candalle, added three more to it, in which he seems to have endeavored to discover every thing that could possibly be thought of respecting the reciprocal relations of the five regular solids. Otherwise, this theory of the regular solids may be compared with old mines, which are abandoned because they cost more than they produce. Geometers will find them at most worth considering as amusement for leisure, or as suggestive of some singular prolem."
What would Kepler have said to this opinion?
As soon as we consider Euclid's work otherwise than as a single
François Foix, Count de Candalle, who died in 1594, in his ninety-second year. He founded a mathematical professorship at Bourdeaux, to be held by persons who should discover a new property of the five regular solids. The first edition of Candalle's Euclid, with a 16th book, appeared in 1566; the second, with 17th and 18th books, in 1578. It is Latin, “Autore Frunc. Flussate Cundalla"
whole, we see at once a necessity for modeling the eight "universally necessary" books into a new manual, of reorganizing it, and accommodating it to a new object. Distinguished mathematicians have undertaken such a remodeling, mostly including as many as possible of Euclid's propositions, and even of his groups of them, in their manuals. But how, it will be asked, can a work, so compactly organized as Euclid's, be capable of being taken to pieces, and its disjecta membra be arranged into a new manual? The explanation is as follows:-Although Euclid set out from one fixed point to reach another, yet he did not proceed in one straight line from one to the other, without any divergence. His single propositions, and still more the groups of them, have a species of independent existence, such that they can be recomposed into new manuals, whose arrangement is wholly different from that of Euclid.
"It is with the fabric of the thoughts
As it is with a weaver's master-piece;
And one stroke affects a thousand combinations."
These expressions of Goethe's Mephistopheles are entirely applicable to Euclid's master-piece.
Shall we now reject these good modern manuals, and use in our mathematical studies the thirteen original books of the "Elements?" Even Kepler, the most thorough-going admirer of Euclid, would object to this. He defended and praised the "Elements" as a magnificent scientific work, but not as a school-book. He would never have recommended our gymnasiasts to study the tenth book, although he charged the celebrated Ramus with having fallen into a grievous error in thinking the book too easy, since it required intellectual exertion to understand it. Montücla, although he expressed himself strongly against a false, enervating, and unscientific mode of teaching mathematics, yet says that geometry must be made intelligible, and that many manuals have subserved this end, which he has gladly made use of in instructing; and that he would recommend the exclusive use of Euclid only to those of remarkable mathematical endowments.
But were Euclid's “Elements" originally a manual for beginners? Shall we compare the learned mathematicians who came from all countries to Alexandria to finish their studies under Euclid, Eratosthenes, or Hipparchus, with gymnasiasts sixteen years old? The Museum at Alexandria was at first, that is in Euclid's time, a mere association of learned men; and only afterward became an educational institu
163 tion.* Euclid therefore wrote his "Elements" for men who came to him already well experienced in mathematical knowledge and exercises. It was because the book was not a school-book that Euclid gave his answer to the king who required him to make geometry easier. But what was the origin of the book?
The reader may perhaps apprehend that this question will lead me into historical obscurity, and obscure hypotheses. But there is no danger.
Montücla says that Euclid, in his book, collected such elementary truths of geometry as had been discovered before him. We know, of at least some of his problems, that they were known before Euclid; such, for instance, as the Pythagorean problems. But, nevertheless, Euclid remains entitled to the credit of having performed a service of incalculable value in the form of the most able and thoroughly artistic editing.
We have already stated the idea which guided him in this task of editing; it was to proceed from the simplest elements, by means of points, lines, and surfaces, to mathematical bodies, and finally to the most beautiful of them, the five regular bodies, and their relations to the cube.
But would geometrical studies, commenced at the very beginning on Euclid's principles, have led immediately to an elementary system such as his? Certainly not. If they would, what occasion would there be for so much admiration of them, and of calling them Elements par excellence, and their author "the Elementarist?"
No man would ever have begun with a point, a non-existent thing, (ens non ens,) and from that proceeded to lines, surfaces, and lastly to solids. Solids would rather be the first objects considered; objects of the natural vision, and the pupil would have proceeded by abstracting from this total idea to the separate consideration of surfaces, which bound solids; lines, which bound surfaces; and lastly of points, which bound lines.
After having proceeded to this ultimate abstraction, to the very elements themselves of the study, Euclid worked out his elementary system as a retrograde course; a reconstruction of solids from their elements. And this reconstruction could only be effected by the aid of precise knowledge and intelligent technical skill; of a full understanding of the laws and relations of figures, solids, &c.
Acute Greek intellects, investigating solids and figures, and subjecting them to actual vision, would of course discover many of their laws at once, and readily. Others, however, could not be perceived by intuition, but could be disclosed to the understanding only at a
* See Klippel, on the Alexandrian Museum, 114, 228.
later period. In examining this cube, for instance, it would appear at once that its sides were equilateral and equiangular; and that one of its horizontal sides was bounded by four vertical ones. But that its edge, diagonal of a side, and axis are to each other as √1: √2: ✓ 3 could not be perceived with the bodily eye, but appears by the help of the Pythagorean problem.
The demonstrations, as is sufficiently evident, must have begun with such as were concrete, simple, and visible, and proceeded to such as were more comprehensive, abstract, and beyond the scope of the senses. For instance, the application of the Pythagorean problem to all right-angled triangles would scarcely have been undertaken at the beginning. But in the case of isosceles right-angled triangles, inspection would show, by a very simple demonstration, that the squares of the sides were together equal to the square of the hypothenuse. If this were proved, the question was then easily suggested, Is it true of all right-angled triangles? If a square were divided by a diagonal into two triangles, it was evident that each of them contained one right angle and two half right angles, the sum of the three being two right angles; and then the question would naturally occur, Is this true of all triangles?
In the same manner it would be necessary to proceed from the simplest and most regular solids and figures to the more complicated and less regular; from those most easily seen by the eye to the more abstract, requiring the use, not of the senses, but of the reason. When at last the most comprehensive demonstration and definition had been learned, there would be no further mention of the previous concrete cases, which had been an introduction to the study of the more abstract ones, but the cases to consider would now be those involved in the definition and demonstration last found.
It has repeatedly been observed that the teacher of a science must adhere to its proper course of development, and must in his instructions follow it more or less strictly. Every pupil ought once to follow this path, which its first discoverers and investigators worked out after
* See my "ABC-Book of Crystallography," (A B C-Buch der Krystallkunde,) pp. IX., XI., XXIII., and 164; and Harnisch, "Manual of the German Common School System," (Handbuch über das deutsche Volkschulwesens,) 1st ed., 1820, p. 232.
+ The demonstration may be somewhat as follows:
A B C, isosceles right-angled triangle. A B D E. the square of its hypothenuse, contains eight small triangles, and the squares on its sides together contain also eight, and all of these small triangles are of the same size and shape.
so many and long-enduring errors, but which the present pupils, with their teacher's aid, now find out in a shorter time, and with certainty.
According to these principles, to which I subscribe, I consider it natural to begin teaching geometry with treating of solids, with which it is highly probable that the actual development of the science began; and to proceed from that point, by abstraction, to the elements. It is here that Euclid's method should be adopted, and that we should proceed by demonstrations, from the elements up to solids. In the former course, it is instruction that leads, and reason silently follows; in the latter, the reason speaks, and the intuition must place faith in it.
Many mathematicians are now agreed that Euclid's demonstrative course of instruction should be preceded by an introduction of an intuitional character. In the theory of forms brought forward by Pestalozzi and his school, in particular, was discovered a preparatory course in geometry, in which intuition was the chief actor, as is the reason in geometry proper.*
Still, however, the beginning was not made with solids, but, in accordance with a method of elementarizing which was pushed even to caricature, with points-unmeasurable, dimensionless points. Lines. come next, and were taught in innumerable and aimless combinations. Lastly, surfaces were discussed; for of solids Schmid's well-known Theory of Form, the predecessor of many more, scarcely spoke at all, and what little was said was not worth mentioning.t
The necessity was afterward felt of beginning with a solid-the cube, for instance; but merely with the design of showing from it the process of abstraction by which to proceed from the solid to the point. As soon as this had been briefly done, they then commonly proceeded to the combination of points, lines, &c., and to other operations, as were just alluded to. How important soever this theory of form may seem to me, and however much I may honor the intelligence, industry, and effort with which this new course of discipline was worked out by able pedagogues, still I can not possibly recognize the method which they followed as the right one.
What I would recommend is, that instruction in geometry should begin, not with such a brief analysis of one or another solid into its geometrical elements, but with a continued study, at some length, of many mathematical solids. And now, if solids are to be both the beginning and the end of the elementary study of geometry, the
Part 2, p. 101.
* Diesterweg Guide," (Wegweiser.) Second edition, part 2, p. 188, &c.
+1 entirely agree with the acute and able judgment passed by Curtmann on the study of Forni in common schools, and on Froebel's " eccentric proposal to use geometrical combinations as a principal amusement for children." See Curtmann's "School and Life,” (Die
Schule und das Leben,) p. 62.