The work of Euclid is preferable, iu our opinion, to any system which has been proposed to supply its place ; simply became the dependence of conclusions upon premises is more distinct than in any other geometrical writing. The defects with which it abounds are trifles which can be remedied as they are met with ; and though there are seldom three propositions together one or other of which will not call for some remark from the teacher, yet such is Euclid, that these very faults, properly noted, are of more value than the greater elegance and more artificial process of less formally rigorous writers. Of the part relating.to PnoPonnos we shall treat under that word.
The necessity of a demonstrative system of arithmetic was practically denied, in this country at least, by the use of a set of dogmatical rules : and this mode of teaching is still too prevalent. Such training would be less prejudicial if the notions of the student on reasoning were more settled, so that he could receive these rules with some fitting idea of their purpose, and of the extent to which they are to be considered as knowledge. As it was, he had no other view of arithmetic presented to him, and his conceptions of number were allowed to be first fettered, and afterwards led astray, by ideas derived sometimes from the recep tion of the conventional for the necessary, sometimes from modifications of truth which are convenient in practice though unsound in theory.
Such an addition to his stock of mathematical power predisposed him to acquiesce in the shifts by which rigour is avoided in other parts of the science ; and an arithmetic from both sequence and accuracy of thought had been excluded was the asylum of geometry against truth, when the latter became itoo difficult for an unenergetic, that is untrained, power of investigation.
We shall conclude this article with some references to works on the history of mathematics, a subject so connected with that of the other sciences, that it would be impossible to say anything on it in few words. The great work of Montucla, continued (from his papers partly) by Lalonde, Histoire des Mathematiques; 4 vols., Paris, 1799 1802, is the most accessible source of information for the reader who desires some detail. The bibliography of Murhard, Litteratur der Mathematischen Wiasenschaften; Leipzig, 1797, will be found a useful accompaniment in the verification of dates. The work of KNstuer, Geschichte der Mathematik,' Gottingen, 1796, is more precise in its account of individual labours than that of Montucla, but does not embrace eo large a field. The works of Bossut, namely, the preface to the mathematical part of the Encyclopedia 316thodique ' and the *Histoire des Mathematiques; 2 vols. 8vo., Paris, 1810, are written in
an interesting manner; and the latter (which was translated by Bonny castle) brings some parts of the history later than its predecessors. Cossali's Origin dell' Algebra,' 2 vols. 4to., Parma, 1799, is an account of the early Italian algebra : its successor in the same subject, "Ilistoire des Sciences Mathematiques en Italie,' Paris, 1838, by M. Libri (of which four volumes have appeared) has gained a character which places it among the first efforts of the kind. Much on the subjects of algebra, logarithms, and trigonometrical tables may be found in Dr. Hutton's Tracts (3 vols. 8vo., 1812), and in the preface of his Logarithms. The work of Montucla, 'Recherches stu. la Quadrature du Cerele; reprinted with additions in 1831, is complete on the subject of which it treats. There is a work on a kindred subject which we have never seen, Reimer, 'Ilistoria Problematis de Cubi Duplicatione; Gottingen, 1798. Delambre's Rapport Historique sur les Progres des Mathematiques depuis 1789; Paris, 1810, is the least specific of his historical works, but, points of nationality apart, is accurate. Professor Peacock's article on arithmetic in the`Encyclopaedia Metropolitana; and his report on Analysis, in the second volume of the Transactions of the British Association,' are full of historical information of the most precise character. Some notices of early English mathematicians, enough to guide the reader to sources of information, are contained in the Companion to the Almanac' for 1837, and in the Magazine of popular Science,' Nos. 18, 20, and 22 (a work completed in 4 vols. 8vo). The eleventh volume of the Memoirs of the Academy of Brussels' contains the Apercu Historique,' &c. of M. Cheeks, a history of geometry and a complete account of its modern progress.
Among works of older date, which are often cited, we may notice Ward's 'Lives of the Gresham Professors,' London, 1760; Heil bronner's Historic Ilatheseos Universe:,' Leipzig, 1742, does not deserve that title, but contains an immense quantity of information on the ancient part ; G. J. Vossius, 'De quatuor Artibus popularibes; Amsterdam, 1650, contains a large number of dates and short biogra phical accounts ; to Blancanus, De Mathematicarum Nature Disser tatio; Bologna, 1615, is appended a chronological list of mathematicians up to the time of Galileo ; the Scolm Mathematicm ' of Peter Ramus, Frankfort, 1627, contains many historical notices. The number of writings which might be referred to as incidentally affording informa tion is of course numerous.