is proved by that circumstance, not to depend on the geometrical, physical or mechanical nature of the elements which are combined, but only upon the mutual relations established between them by addition, subtraction, division, or any other operation of calculation. It results from this, that if in each kind of quantities, which are combined, whether they be lines, surfaces, solids or masses, we choose one of them at pleasure to serve as a unit of its kind, all the others which it is necessary afterwards to combine either with each other, or with the unit itself, are only collections of the original unit; so that all the calculable relations to which we subject them, become in truth numerical problems. This is the reason why algebra is applied to them always in the same manner, whatever may be the absolute nature of the quantities so compared.

The first step to be taken to apply algebra to the resolution of the problems of linear geometry, must, therefore, be, to fix on a particular length of line which is to be used as the unit of of all other linear dimensions. Then, all these lines will be represented by numbers entire or fractional, rational or irrational, and we may perform upon them all the operations of arithmetic. In this way, we may conceive lines added to, or subtracted from each other, multiplied into, or divided by each other; and this is the only point of view under which we can understand the meaning of such operations when they are performed upon lines. This method of proceeding will enable us, mutatis mutandis, to express and combine every species of quantity whatever, and to subject them to all the operations which are necessary to arrive at a desired result.

Our author first treats of the geometrical construction of algebraic quantities. To this end, he makes use of expressions made up for the occasion, instead of the more natural and effectual method of giving an algebraical solution to geometrical problems, and then explaining how a construction in geometry may be substituted in place of the numerical solutions to which students in the preceding part of their course had been accustomed. The style, however, in which he presents the subject, is clear, a valuable quality in all Bezout's writings. The next thirty-two pages are entirely taken up in the solution of problems, and upon these and upon the manner of their solution, we shall make some desultory remarks. As in algebra, so here, no certain rule can be given for putting a problem into an equation; but the difficulties in both branches, though similar, are not equal, a problem in geometry being in general less easy

to put into an equation than an ordinary problem in algebra. In the latter, it is most commonly sufficient to translate by the aid of algebraic signs, the expressed conditions of the enunciation; or if not, the implied conditions which are easily deducible from them. Besides, the given and the unknown quantities in algebraic problems are evident upon mere inspection; while in a geometrical problem, which is almost always reduced in the last resort, to determining the position of one or of several points, much attention and sagacity are frequently necessary to determine the nature of the relations, which, when algebraically expressed, will lead to a simple and elegant construction of the problem. It is true that it is always easy to find in the figure which the enunciation suggests, and with the aid of the constructions which naturally present themselves, a first essay at solution, by recurring to the principal relations of geometry, such as the properties of right-angled triangles, of similar triangles, or of lines in and about the circle. But that which requires special address;—that which constitutes particularly the art of the analyst, is, to discover the most direct course by whichto pass from the known to the unknown quantities, and to select among all the relations which connect them, those which are most suitable for calculation, and to fix on constructions capable of leading to simple equations and elegant results.

In the course of the solution of these problems, the interpretation of negative results is given, a subject which has been considered abstruse and perplexing. The principal difficulty appears to us to have arisen from confounding the mathematical relations, which in their nature are as permanent as the universe itself, with the language which has been invented with a view to investigate these immutable relations, and which is entirely matter of convention. The laws of algebraic combination sometimes lead to results which are of difficult interpretation, but however the case may be, it is certain that fact and the dictates of common sense should not be violated in giving them a signification. Chiefly after Bourdon, we shall attempt a summary of the rules which respect the interpretation of negative results.

1st. The sign sometimes indicates, as in algebra,* that the enunciation of the question requires to be changed in certain respects.

2d. It happens sometimes that the equations of a problem give, with respect to signs, a number of results, of which a single Algèbre, par Lacroix, Paris, 1818, p. 88.

one only is capable of satisfying the enunciation; the others are solutions of other problems which have a relation more or less intimate with the proposed question. The difficulty consists then, in discovering among these different expressions those which refer to the question itself, and those which are foreign to it, or which refer to it indirectly.

3d. As often as in the resolution of a problem, the unknown quantity represents the distance from one fixed point to another reckoned upon a fixed right line, and we obtain for the expression of this unknown quantity, results, some of which are positive and others negative; if it is agreed to reckon the positive values in one direction departing from a fixed point, the negative values must be reckoned in the opposite direction. This rule is the same with that which we formerly applied to the different trigonometrical lines in the circle.

4th. We may always make negative solutions disappear, by referring the point sought to another fixed point, whose distance from the first fixed point is sufficiently great to assure us that all the points capable of satisfying the enunciation, will be on the same side with respect to this second point, and this is always possible since the line upon which these distances are reckoned, may be indefinitely produced. Negative results arise entirely from this circumstance, that the origin of the distances was at first chosen in a position intermediate between the points sought; and the sign indicates the difference of position of these points with reference to the first fixed point.

5th. If, in the resolution of a question, whether it be a problem or a theorem, we wish to take into view, distances between a first fixed point and other points situated with this upon the same line but in different directions, and we regard as positive the distances reckoned in one direction, we must regard as negative those which are reckoned in the opposite direction.

In the solution of problems, every thing depends on a happy selection of unknown quantities. A remarkable instance of this is seen in a problem given in Lacroix's Application de l'Algèbre a la Géométrie, p. 106, taken originally from Newton's Arithmetica Universalis. To be understood, however, the solution must be read, as it does not admit of being represented. To those who are desirous of perfecting their skill in the solution of questions, we can recommend nothing superior to Newton's treatise just mentioned, and Carnot's Geometrie de Position.

The problems of which we have hitherto spoken, are of the kind called determinate, because the unknown quantity is susceptible of but a finite number of values, but algebra applied to

geometry would be of small comparative importance, if the sphere of its operations were thus limited. The chief excellence of the method is not seen, and its power of expression is scarcely felt, until we come to apply it to the researches of indeterminate geometry.

We may consider all lines, whether right or curved, as susceptible of being represented by equations between two variables; and, reciprocally, any equation between two indeterminates may be interpreted geometrically, and may be considered as representing some line, all the points of which in succession, it can furnish the means of tracing. Lacroix represents all the the conic sections under the general formula ;

Ay2+Bxy+Cx2+ Dy + Ex= F.

This method, so fertile in consequences, in the hands of modern analysts, that it has changed the whole face of mathematics,* may even be generalized so as to apply to equations with three variables, which represent surfaces as is the case in the treatises of Lacroix and Biot.

The two branches of the application of algebra to geometry, which are here brought into view, to wit, the one limited to determinate, and the other embracing indeterminate geometry, are not only distinct in their nature and object, but they are distinguished in the history of the mathematics. The invention of the last branch belongs to the celebrated Des Cartes. Before his time, algebra had only been applied to determinate problems of geometry. The first applications of this kind, had even been simply numerical, being limited to finding and calculating arithmetically the numerical value of the unknown quantities, according to their final algebraic expression. Towards the end of the fifteenth century, Vieta, a celebrated French analyst, thought of representing these expressions by geometrical constructions. These constructions, however, were inadequate to interpret the values of the unknown quantities in the case of indeterminate equations. Des Cartes made an immense advance, by shewing that such equations represent geometrical loci, and it is scarcely too much to say, that by this discovery, he created the application of algebra to geometry, the constructions of which, in the hands of Vieta, were confined to a particular class of problems. Any problem of geometry is always reduced to finding a certain number of points, lines or surfaces, the position or configuration of which may satisfy certain given conditions. We may even consider the investigation of points as a problem

* Lacroix, Application, &c. p. 116.

of the intersection of lines. If we have a general method of finding the equation of lines according to the enunciation of the geometrical conditions which they are required to satisfy; and, reciprocally, if we are able to discover the form as well as the course of the lines, when the analytical equation which expresses them is given, there will be no problem of geometry, however complicated, which we shall not be able at once to write algebraically, and in this way to reduce to a combination of purely analytical equations. It was by the aid of this secret, that Des Cartes, at the age of twenty years, passing through Europe in the simple style of a soldier of fortune, resolved at sight and as matter of amusement, all the geometrical problems which the mathematicians of different countries were in the habit of sending to each other by way of public challenge, according to the custom of that period.* In 1617, while in the Dutch army under Prince Maurice, being quartered at Breda, some one had affixed on the corner of the street a mathematical problem, requiring the solution of it. Des Cartes observing several persons reading this card, which was in Flemish, requested one of them to translate it into Latin. The person addressed at once complied, but imposed the condition that he should send him the solution of the problem. The air of Des Cartes in accepting the condition was so determined as to excite the surprise of the other party, who could scarcely believe that a young officer could solve a problem so difficult. From the card which he received, Des Cartes learned that he had been conversing with Isaac Beeckman, the Principal of the College of Dordrecht. The next day he went to Beeckman's house with the problem solved, and shewed him the construction of it. In consequence of this singular interview, they became correspondents and friends during the remainder of their lives.

Returning from this digression, we have a few more words to say respecting Bezout's treatise and its translator. The remainder of the volume is occupied in discussing the conic sections, the most obvious properties of which are demonstrated in a style which has little to recommend it, except that the learner will follow it without difficulty.

From the circumstance that the Cambridge course is made up from several writers, there is not that uniformity in the style and in the use of terms which it is desirable to maintain. In the algebra, p. 31, an objection is made to the use of the term dimensions, while it occurs frequently in this voluume. In p.

* Biot, Essai de Geom. Anal. p. 75.

+ Vide p. 72.