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by James Williamson, Oxford, 1781, in 2 vols. 4to. The translation is here as literal as any authorized version of the Bible; and, in like manner, the additional words of English necessary to complete the sense are inserted in italics.
As to the editors who amend to their fancy, and then say, this must be what Euclid wrote, we have of course nothing to do with them, writing as we now are upon evidence and evidence only, and being exceedingly dubious of the fact that Euclid, any more than Thucydides, wrote otherwise than as it is set down that he wrote in the remaining manuscripts. If these be corrupt let them be restored, if possible, by context, by comparison, or by good conjecture within the most approved canons of criticism. If, after all, the Alexandrian Greek will not do to teach geometry by (which is quite another question) let him be amended or abandoned, but let not such amendments be called Euclid. Robert Simson producing that which he thinks best, in the way of addition, alteration, or comment, is not only bearable, but admirable ; Robert Simson declaring that whatever he thought Euclid should have written, must be that which Euclid did write, is a false critic, and a teacher of falsehood, though of course not intentionally; Robert Simson declaring that he had discovered, by reflection, words and sentences of Euclid which had been buried in oblivion for ages, was not one whit less absurd than the discoverers of hidden treasures by the divining rod : and those who printed Robert Simson's notes in school Euclids, were guilty of great inconsistency, unless they could excuse themselves by saying they intended to destroy any notions of sound criticism which a youth might acquire from the notes to his classical authors, by the perusal of those attached to his mathematical guide.
It is much to be regretted that the solid initiation which Euclid enables the student to obtain, is beginning to be abandoned ; and if there be one thing more than another which the friends of liberal education should bestir themselves upon, it is the defence of this unequalled system. “Lagrange,” says Peyrard, “ often repeated to me that geometry was a dead language; and that he who did not study geometry in Euclid, did exactly like one who learnt Greek and Latin by reading modern works written in those languages.” We may trace the consequences of the abandonment of Euclid in the general state of elementary writing in every country in which it has been abandoned. Ålgebra, left to the habits which it forms without geometry, always grows lax in its reasonings; and those who
have lost Euclid, have always formed a less rigorous system. If we could find any tendency to deny these assertions, we might argue the grounds on which such denial was made: but no one pretends to show the substitute for Euclid; no one professes that algebra* is everywhere of equal rigour. Some desire mathematics only as an instrument in the investigations of physics : let them have their approximative system, by all means; but we are now speaking to those who think of the formation of the mind to the utmost exactness of which it is capable, and who see clearly that it has pleased God that the higher and finer parts of civilization should be much advanced by the cultivation of critical accuracy in all things in which it is attainable. To be brought by degrees to the keenest perception of truth and falsehood, is the highest intellectual hope
Now in this process there is, so far as mathematics are concerned, no commencement like Euclid; a writer who seized realities, separated the necessary characters from all that was artificial or conventional, and took the ground on which the beginner could appreciate what he was doing, in a manner which never was equalled, and probably never will be. When we look at his rude, but certain, mode of exhibiting to the young mind, not yet prepared for the nicest distinctions, the raw material of its own conceptions, and using it in a manner which obtains such an instantaneous and intuitive assent as never could be given before to anything in which there was progression from one idea to another, we think we see that mind first feeling its own feet, and learning the possibility of walking alone. Its faint and tottering steps may indeed need the strong support of which it is conscious, but there is a hardness in the ground, and a success in each successive step, which gives an increasing confidence in the future. Many and many a student, mystified by algebra, as taught in its principles-amused to contempt by a science of which (to him) the subject-matter is all conundrum about apple-women, who tell each other how many apples they have got in language which needs an equation ; and men who buy flocks of sheep at prices which can only be told by completion of a square and extraction of a root-many such students, we say, have only their Euclid to give them any idea of what real science is : that is, at the commencement of their career. They may afterwards find algebra to be what could not have been guessed from equation books; but were it not for what they see from the beginning in geometry, they would have no encouragement to hope for either light or knowledge, from the first year's study.
* It may be hoped that algebra will be thoroughly rigorized by the views which have lately been promulgated; but the time may be distant at which these views can be made the elementary foundation of the subject; and even then, it may be found that its abstract nature requires a strength of mind previously drawn from geometry.
Independently of the positive superiority of Euclid, there is a strong reason for retaining his system, drawn from the frailty of humanity. There is no reasonable prospect of retaining sound demonstration if Euclid be now abandoned; for it is evident that such abandonment as has been made, has arisen from a disposition to like easy laxness better than difficult rigour. We will not speculate upon what might be substituted for the Elements, when we have reason to know what would be substituted: the former question may be adjourned until the advocates of change show themselves to be really actuated by a love, not of scientific results, but of scientific truth. As long as Euclid is in request, be it only by a minority, the majority are ashamed of more than a certain amount of departure from soundness: but the direction of that departure shows clearly enough what would take place, if, instead of merely retiring into the darker places, the algebraists were allowed to put out the light altogether. There is not a better work, next after Euclid, than the Geometry of Legendre; which, when the dangerous elements are past, has an elegance unknown in Euclid himself. But, considered as an exposition of geometrical principles, it is hardly worth a passing notice: the first books are a mixture of arithmetic and geometry, in which the province of the two sciences is confounded, or they are made, in all points of real difficulty, to darken each other; while Euclid, by keeping them distinct till the proper time, has made each help the other. In Legendre, the horse and foot are in alternate ranks, instead of separate regiments; and one part of the service is always either cramping the movements of the other, or getting tripped up by it. When the two arms are likely to quarrel, a general order comes from head-quarters in the shape of a supposition, or an imagination : " par exemple, si A, B, C, D, sont des lignes, on peut imaginer qu'une de ces quatre lignes, ou une cinquième, si l'on veut, serve à toutes de commune mesure.” (Book III., note on the definitions.) How nice! Legendre knew as well as any body that there are abundance of cases in which lines have no common measure : then, says he, you must imagine a line which serves as a common measure to them all, a sort of acting common measure, which does the duties, and receives the pay and appointments, under a commission signed by the imagination. Euclid, stupid Euclid, had no imagination. The stark staring nonsense which we have quoted, and which can only be treated with ridicule, is but a sample of what we may expect, if we abandon what we have, before we have received something better. Lacroix, to whom elementary writing, in everything but geometry, is more indebted than
, to any other man living, does not proceed quite so absurdly; but he only escapes at the expense of declaring geometry to be an approximate science. He proves that a common measure may be found with an error imperceptible to the senses, and on such a common measure he founds his geometry. Let such ideas take clear possession of the field, and we should soon come to this—that algebra would be held perfectly sufficient, and that all which is necessary at the outset might be proved by a ruler and compasses, or by an imagination, according to the taste of the learner; nay, even an act of parliament would perhaps be thought sufficient.
The senate of the University of London (not what was the University of London, now University College, but the body which was chartered in 1837) in the announcement of the qualifications required from candidates for the degree of B.A., specifies the following amount of knowledge in geometry: the first book of Euclid—the principal properties of triangles, squares, and parallelograms, treated geometrically, the principal properties of the circle, treated geometrically—the relations of similar figures—the eleventh book of Euclid to Prop. 21. We do not think this attempt to abandon Euclid
. a particularly happy one. The first article seems to be a concession to true geometry, by way of compliment to the vigorous growth which it has heretofore gained in our country. The second might be mended in two ways; squares
parallelograms looks like Londoners and Englishmen, or cats and animals, while treated geometrically is a puzzle. Does it mean that a young student, who must learn the first book of Euclid, is at liberty to deduce the properties of squares and parallelograms which he does not find there, in any way which he pleases, from any other system? The same question may be asked of what are called the principal properties of the circle; and if the answer be in the affirmative, we cannot but wish the new University would have taken a page out of the book of the old ones; while if it be in the negative, we may well ask, why it was not simply required that the candidates should have studied the first four books of Euclid? Next come the relations of similar figures, no doctrine of proportion being mentioned except what in a preceding part of the same list is called algebraical proportion. Here again a doubt arises, as to what is to be learnt: will it do if a student come with Legendre's acting common measure, or Lacroix's tiny errors qui échappent aux sens par leur petitesse? These are questions which many of the well-educated portion of the community will ask themselves before they make up their minds to think the B.A. degree of the London University a worthy object of ambition for their sons: these are questions which the enemies of the liberal cause will answer their own way in their own minds: they will turn to the ancient institutions, which, whatever may have been their faults and their prejudices, have kept the ark of liberal knowledge among us through centuries upon centuries, and will say with a smile, and what is worse, will be justified by the event, that the London University will be a mother of learning when Oxford and Cambridge are defunct—but not till then. Hoping for a better result, we trust that the day is not distant when methods will appear of more importance than mere matters of conclusion to those who guide the new institution: a very few years will point out the working of the present chequered scheme.
We shall now turn our attention to one point of the text of Euclid on which lawless alteration has been perpetrated, in what are called the axioms. Euclid distinguishes three preliminaries to geometrical discussion: definitions, in which he is not metaphysically anxious to satisfy any canon of definition, but only to be very sure that his learner shall understand of what things his words speak;* postulates (airhuara), demands upon the sense of the reader, without which he professes to be unable to proceed to reason on the properties of space; kovai čvvorai, common notions, matters of intuitive assent, which are either common to all men, or common to all sciences (most probably the former; if the latter, the question about to be discussed need not be entered upon), which must be granted, because it is matter of experience that all men do grant them, even those who never heard of geometry. The
* All the objections made to Euclid's definitions, distinctly show that the objectors knew what Euclid meant: that is, that so far as they were concerned the definitions were good.