ples of the others." But this property defines these magnitudes as proportionals. In Playfair's edition an error has slipt into this proposition, the words, "any whatever," and “any” which occur, are transposed; though the argument is correct.*

d. Property from property.-Ex.-If a side of a triangle be bisected by a straight line parallel to the base, the other side is bisected also. Proportionals are obtained connecting the magnitudes which possess the given property with others, which latter, therefore, themselves possess the property. There is, however, great variety in this application of the doctrine. The reasoning is generally of the nature of a reductio ad absurdum disguised.

Let us refer to the second case of Prop. 2, B. vi. of Euclid. Here it is shown that two triangles are equal to one another, viz. BDE to CDE, without apparently applying congruity at all. On examining the process we shall find that the demonstration has been effected by taking equimultiples of the two triangles, and showing that they always lie together on the same side of the multiple of a third ADE, and thence that the two triangles cannot be other than equal. This example, if carefully studied, will do much towards communicating a correct

*It is a curious fact that confusion of this essential distinction exists in all the best editions of the Greek text. In every one which I have seen, the words Tvx, any whatever, are more or less misplaced. Simson pointed out the circumstance respecting Briggs's and Gregory's editions. I find the same thing in the Basle edition, (1533). But what is more remarkable, Peyrard's edition, which supplies the omission of these words in the construction, from the Vatican MS. (190), retains the same words improperly in the demonstration, in speaking of K, A, M, N being equimultiples of A, B, г, A. Camerer follows Peyrard in this place. It is singular indeed, if not one of the MSS. is correct.

conception of the method. It is not congruity of the magnitudes that we establish directly, but the impossibility of the want of congruity of their equimultiples.

In conclusion, it is requisite to say a few words on unequal ratios. The definition given by Euclid (Simson's Def. 7) is open to the objection,*" that by it the ratio of A to C might, at the same time, be both greater and less than that of B to C." The ground of the objection is this: it is possible to conclude that one particular multiple of the first exceeds a number of times the second, whilst the same multiple of the third does not exceed the like number of times the fourth; and that another particular multiple of the first falls short of a number of times the second, whilst the same multiple of the third does not fall short of the like number of times the fourth. And, assuredly, although this is really impossible, it is inconsistent with strict geometry to assume the fact in framing a definition. The difficulty would be removed by defining unequal ratios simply, and deducing as a proposition the existence of the species of inequality, greater and less. What is required to be proved may be enunciated in the following way: If four magnitudes be such that any one multiple of the first is greater than a certain number of times the second, whilst the same multiple of the third is not greater than the like number of times the fourth; then if any multiple whatever of the first be not greater than a number of times the second, the same multiple of the third is less than the like number of times the fourth. The demonstration of this proposition may be effected without difficulty.

* T. Simpson, Elements of Geometry. Note.

LECTURE V.

ON THE FOUNDATION OF ALGEBRA.

History-want of attention to the principles amongst many writers-exceptions, Peacock, Hamilton, De Morgan-my own views-definition of algebra-a conception of the nature of the symbols and operations requisite geometrical illustrations-imaginary geometry, and Reid's geometry of visibles (note)-reasons for requiring an idea of operations -algebraic definitions of indices-signs-imaginary quantities—arguments in support of the arithmetical derivation of algebra-difference of two systems-reasons for preferring the arithmetical derivation-1° in the other system risk is run that the conditions be inconsistent with each other-2° a direct and reciprocal law have to be applied simultaneously-3° the transition to arithmetic as difficult as the transition from it-conclusion.

IN the notational sciences the civilized nations of antiquity made little progress. The arithmetic of Greece and Rome was deficient in many of the most important characteristics of our own. Owing, probably, to the want of a symbol to denote zero, the rudest method of representing large numbers was for ages current amongst a people who left no other branch of knowledge uncultivated and unimproved. The real arithmetic of the Greeks was indeed not a notational science. It afforded no facilities for operating on numbers, and hardly sufficed, in the most inartificial manner, to represent them.* Nor was their algebra of a much higher character. I do not allude to the fancies of the Pythagoreans, but to the more

* Consult Wallis, Algebra. Leslie, Philosophy of Arithmetic, and Art. Arith. in the Encyc. Metrop.

important discoveries of Diophantus, who wrote somewhere about the fourth century. His treatise is nearly destitute of general symbols, or of anything which could mark a formal science. In its constitution it is some. what similar to the Indian* algebra, but greatly inferior to it in the nature of its processes. At the revival of letters, algebra was imported from Arabia into Italy by Leonardo di Pisa.† It does not appear that the science spread to the adjacent countries for a considerable time: Neither did it make much progress in Italy. From the work of Lucas de Borgo, written at the end of the fifteenth century, we find that nothing was known beyond quadratic equations, and that no systematic notation had been invented at that time. In the sixteenth century algebra assumed a higher character in the hands of Ferreus, Cardan, Tartaglia, and Ferrari in Italy, and Vieta in France. To the Italian school we owe the solution of cubic equations, and to Vieta many important additions to the science. Previous to his time the unknown quantities only had been designated by letters, he adopted a similar notation for those which are known, and thus communicated to his solutions a generality which is lost in mere arithmetical computation. About the same time flourished Michael Stifels, the father of the science in Germany, in whose writings we discover the germ of the invention of logarithms. Nor were our own countrymen backwards.

Recorde, and especially Harriot did much towards the advancement of the science by systematizing, which was

* Hutton's Tracts, v. ii. and Colebrooke's Indian Algebra.

+ Cossali Istoria d'Algebra.

at that time of the utmost utility. We cannot stop to give a detailed account of the discoveries of Des Cartes, Lucas Valerius, Peletarius, and others. Suffice it to mention the names of Napier, to whom every science is indebted for his invention of logarithms, Briggs, Ramus, Nonius, Bombelli, Stevinus, Albert Girard, Oughtred, Pell, Wallis, Fermat, Newton, Leibnitz.* In later times the science of algebra has been advanced, and many difficulties have been overcome by Euler,† Maseres, Waring, Lagrange, Budan, Gauss, Horner, Abel, Jacobi, Cauchy,

* Consult Hutton's Tracts, Vol. ii.

+ Euler, Elements of Algebra, &c. Waring, Meditationes algebraicæ, &c. 5 vols. 4°. Camb. Maseres, on the negative sign, 1758. Scriptures logarithmici, 6 vols. 4°. Tracts on the solution of equations, 1803. Lagrange, Traité de la résolution des équations numériques. Budan, Nouvelle méthode pour la résolution des équations numeriques, 1807. Gauss, Disquisitiones arithmetica, 1801. Theoria biquadricorum, &c. Horner, Philos. Trans. 1819. Abel and Jacobi in Crelle's Journal. Cauchy, Analyse algebrique, 1821, Exercises, &c. Journal de l'Ecole polytechnique, passim. Gompertz, on imaginary quantities, 1818. Warren, on the square root of negative quantities, 1828. Peacock, Algebra, 1830. Fourier, Analyse des équations détérminées, 1831. Sturm, Mémoires des savans Etrangers, 1835. Graves, Trans. Roy. Soc. 1829, &c. Sir W. Hamilton, Trans. Roy. Ir. Ac. v. xvii. and Report of British Assoc. Vol. v. Murphy, Trans. Camb. Ph. Soc. Vol. iv. &c. Roy. Soc. 1837. Theory of equations, 1839. De Morgan, Tr. Camb. Ph. Soc. Vol. vii. Elements of Algebra.

I am aware that to leave the impression that this list embraces all the discoverers in algebraic science in modern days would be to do injustice to many, especially of our own country. It is want of space, and not want of inclination, which compels me to omit a reference to the writings of Ivory, Holdred, Jerrard, and a host of others. Ample information will be found in Prof. Peacock's Report, and in Young's excellent Theory of Equations, 2d edition. The journals of Gergonne, Crelle, Liouville, the Ecole Polytechnique, &c. will supply the history of continental discoveries.