In the year 1655, the first Lucasian Professor of Mathematics, Isaac Barrow, M.A., Fellow, and afterwards Master of Trinity College, published a Latin translation of Euclid's Elements; and five years afterwards an English translation, of which more than one Edition was printed; the latest bears the date of 1751. A new Edition of Dr. Barrow's Mathematical writings, edited by Dr. Whewell, has recently been printed at the University Press.

In the year 1756, Dr. Robert Simson, when he had been Professor of Mathematics in the University of Glasgow for upwards of forty years, published a Latin and also an English translation of the first Six and the Eleventh and Twelfth books of Euclid's Elements. Dr. Simson's new translations of the Elements were favourably received at Oxford, and the Rev. Robert Smith, D.D., then Master of Trinity College, Cambridge, entertained so high an opinion of their superiority, that he earnestly promoted the introduction of them at Trinity College. There is extant a letter in the hand-writing of Dr. Simson, in which he refers to thirty-six copies subscribed for by the Rev. Dr. Smith at Cambridge; and two letters of Dr. Smith, from which it appears, he paid to Dr. Simson's publisher in London forty pounds within a few shillings, for copies of the work which he had received. Dr. Smith was also the founder of two annual prizes which bear his name, for the two most distinguished proficients in Mathematics and Natural Philosophy among the commencing Bachelors of Arts in the University.

Dr. Simson's translation having been thus introduced by Dr. Smith at Trinity College, it appears to have been preferred in the University, until at length it became established, as the authorised text of the English Version of Euclid's Elements of Geometry. Time, usage and experience have concurred to confirm this estimation. The translation, though not faultless, having now maintained its character for superiority at the English Universities for more than a century, is entitled, at least to the careful and cautious consideration of any editor in making amendments and alterations. Some years ago Simson's Euclid was published in "the Symbolical form", and became extensively used both at Cambridge and elsewhere. This form is open to very serious objections. A more correct taste has of late years returned, and the Symbolical form of Euclid's Elements is not deemed at Cambridge, the most appropriate for introducing the

student to a correct knowledge of the Science of Pure Geometry. It may be added, that this opinion has been publicly recognised in the University. In the Examination papers in Pure Geometry both in the Senate-House and in College examinations, a caution has, of late years, been printed at the head of the papers against the use of Algebraical Symbols in writing the demonstrations of Euclid's propositions. There is still among mathematicians a difference of opinion with respect to the desirableness of exhibiting the demonstrations of Pure Geometry in Algebraical language. It is by some urged that the symbolical demonstrations are shorter and save time; by others it is replied, that 66 attempts at abbreviation have caused endless confusion". It may perhaps be granted, that when a learner understands clearly the distinction between Pure Geometry and Algebraic Geometry, a certain latitude may be admitted in the use of symbolical language in Pure Geometry, which if allowed at an early stage of his progress, would most probably have led him into error, by the interchange of the terms of one Science with the written notation of the other.

A distinguished Geometer of the last century has thus fairly stated one side of the question :-"The frequent use of symbols, common to the algebraic notation, may perhaps be looked upon as repugnant to the rigour and strictness of Geometry. But it is not the use of Symbols (which some more scrupulous than discerning, have condemned) but the ideas annexed to them, that render the consideration Geometrical or un-Geometrical. In Pure Geometry, regard is always had to absolute quantity of some one of the three kinds of extension, abstractedly considered; and whatever symbols are used here, are to be considered as expressive of the quantities themselves, and not as any measures, or numerical values of them. Thus, by A× B taken in a Geometrical sense, we have an idea, not of the product of two numbers (as in the Algebraic Notation,) but of a real, rectangular space, comprehended under two right lines, represented by A and B, and two others equal to them. So likewise, is not to be understood here in the light of an algebraic fraction, but as a right line which is a fourth proportional to three other right lines, represented by A, B and C. These distinctions are absolutely necessary to those who would have an accurate idea of the subject."

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Professor De Morgan, on the other side maintains, that "Those who introduce Algebraical symbols into Elementary Geometry,

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destroy the peculiar character of the latter to every student who has any mechanical associations connected with those symbols; that is, to every student who has previously used them in ordinary Algebra. Geometrical reasoning, and Arithmetical process, have each its own office: to mix the two in elementary instruction, is injurious to the proper acquisition of both."

The following opinion of Sir Isaac Newton will be regarded as not without weight on this point: "Equations are expressions of Arithmetical computation, and properly have no place in Geometry, except as far as quantities truly Geometrical (that is lines, surfaces, solids, and proportions) may be said to be some equal to others. Multiplications, Divisions, and such sort of computations, are newly received into Geometry, and that unwarily, and contrary to the first design of this Science. For whosoever considers the construction of Problems by a right line and a circle, found out by the first Geometricians, will easily perceive that Geometry was invented that we might expeditiously avoid, by drawing lines, the tediousness of computation. Therefore these two Sciences ought not to be confounded. The ancients did so industriously distinguish them from one another, that they never introduced Arithmetical terms into Geometry. And the moderns, by confounding both, have lost the simplicity in which all the elegance of Geometry consists. Wherefore that is arithmetically more. simple which is determined by the more simple Equations; but that is geometrically more simple which is determined by the more simple drawing of lines; and in Geometry, that ought to be reckoned best which is geometrically most simple." The judgment of Sir Isaac Newton therefore is decided and clear, that the two Sciences of Pure Geometry and Algebraic Geometry ought not to be confounded. The remark of La Place is also worth notice:-"Cependant, les considérations géométriques ne doivent point être abandonnées; elles sont de la plus grande utilité dans les arts. D'ailleurs, il est curieux de se figurer dans l'espace les divers résultats de l'analyse; et réciproquement, de lire toutes les affections des lignes et des surfaces, et toutes les variations du mouvement des corps, dans les equations qui les expriment. Ce rapprochement de la géométrie et de l'analyse répand un nouveau jour sur ces deux sciences; les opérations intellectuelles de celle-ci, rendues sensibles par les images de la première, sont plus faciles à saisir, plus intéressantes à suivre."

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At the present day, the school studies of Elementary Geometry and Arithmetic occupy the time of students at the University which ought to be applied to studies, which with greater propriety may be termed Academical. The University of Cambridge by its recent legislation has attempted to enlarge and extend the sphere of Academical study by the institution of the Natural Science, and the Moral Science Triposes, and that of Constitutional History and Law. The limited success which has followed these laudable measures, may be mainly, if not entirely attributed to the imperfect and otherwise defective elementary knowledge with which a large number of students commence residence in the University. It is essential that those students who desire to secure the advantages of a course of study at Cambridge, should have previously mastered the Elements of Geometry and Arithmetic, as well as those of the Classical languages. Indeed it would be no detriment to the character of the University of Cambridge, if it were to adopt the restriction (μηδεὶς ἀγεωμέτρητος εἰσίτω) which is reported to have been maintained by Plato with respect to his disciples before they were admitted to pursue the more advanced intellectual studies. The Ancient Statutes of the University of Cambridge, set forth by Edward VI. ordained,——“ that a student's first year should be devoted to Arithmetic, Geometry, and as much as he could manage of Astronomy and Cosmography." At this period, students were considerably younger at the time of their admission than at present, and at the end of their first year, the studies of Geometry and Arithmetic were mastered and completed, before they had actually attained the age at which students now commence their residence at the University. At a later period in the history of the University, however, when Geometry and other elementary subjects had fallen into neglect, it was found necessary to remind the Candidates for Academical honours, that a competent knowledge of Euclid's Elements was essential, as will be seen from the following caution published by the Vice-Chancellor on May 20, 1774:-"That unless a person be found to have a competent knowledge of Euclid's Elements, and of the plainer parts of the four branches of Natural Philosophy, no attention will be paid to his other Mathematical knowledge. And that in every branch of science, the clearest and most accurate knowledge, rather than the most extensive, will be regarded as the best claim to Academical Honors."

Before closing these remarks, the Editor desires to direct the

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student's attention to the opinions of two distinguished men, on the utility and importance of the Geometrical Element in Mathematical Studies. The late Rev. Dr. Chalmers has stated:-"I am not aware that as an expounder to the people of the lessons of the Gospel, I am much the better for knowing that the three angles of a triangle are equal to two right angles; or that the square of the hypotenuse is equal to the squares of the two containing sides in a right-angled triangle. But I have a strong persuasion, that both the power to apprehend and the power to convince, may be mightily strengthened-that the habit of clear and consecutive reasoning, may be firmly established by the successive journeys which the mind is called on to perform along the pathway of Geometrical Demonstration. The truth is, that, as a preparative, whether for the bar or for the pulpit, I have more value in Mathematics for the exercise which the mind takes as it travels along the road, than for all the spoil which it gathers at the landing-place". And Dr. Whewell's opinion is to the same effect: "Yet perhaps it may sometimes appear, both to teachers and to students, that it is a waste of time and a perverseness of judgment to adhere to the ancient kinds of Mathematics [Arithmetic and Geometry], when we have, in the modern Analysis, an instrument of greater power and range for the solution of Problems; giving us the old results by more compendious methods; an instrument, too, in itself admirable for its beauty and generality. But to this we reply, that we require our Permanent Mathematical Studies, not as an instrument, but as an exercise of the intellectual powers; that it is not for their results, but for the intellectual habits which they generate, that such studies are pursued. To this we may add, as we have already stated, that in most minds, the significance of Analytical Methods is never fully understood, except when a foundation has been laid in Geometrical Studies. There is no more a waste of time in studying Geometry before we proceed to solve questions by the Differential Calculus, than there is a waste of time in making ourselves acquainted with the grammar of a language before we try to read its philosophical or poetical Literature."


October 1, 1861.

R. P.

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