ADVERTISEMENT. LACROIX'S ALGEBRA has been in use in the French schools for a considerable time. It has been approved by the best judges, and been generally preferred to the other elementary treatises, which abound in France. The following translation is from the eleventh edition, printed at Paris in 1815. No alteration has been made from the original, except to substitute English instead of French measures in the questions, where it was thought necessary. When there has been an occasion to add a note by way of illustration, the reference is made by a letter or an obelisk, the author's being always distinguished by an asterisk. In a review of the two first parts of the Cambridge course of Mathematics, which appeared in the American Journal of Science and the Arts for 1822, after many favorable remarks, the writer, speaking of Lacroix's Algebra, observes, that "there are instances of incorrect translation at pages 18, 23, 54." It is regretted that the passages referred to were not more particularly pointed out. The places mentioned however, have been carefully examined and compared with the original. At page 18 the only passage to which the above remark can be supposed to apply, is the following; "and by arranging the letters in alphabetical order, they are more easily read;" of which the original reads thus: "et en intervertissant l'ordre des multiplications pour conserver l'ordre alphabétique, plus facile dans l'énonciation des lettres." Here, as in other parts, a little latitude is used for the sake of perspicuity, and of preserving the English idiom; but it is presumed that the sense is fully and exactly rendered. At page 23 there was clearly a mistake, the sense being the reverse of that of the orginal, and of that which the connexion obviously requires. At page 54, the only inaccuracy to be found is in printing "multiplier" for "multiple." "At page 37" [97], says the reviewer, "the last clause, and retaining the accents which belonged to the coefficients,' does not express the meaning of the original." The original of the whole passage runs thus ; "en changeant le coefficient de l'inconnue qu'on cherche, dans le terme tout connu, et en conservant d'ailleurs les accens tels qu'ils sont." It is not easy to perceive in what the defect of the translation consists. A literal rendering would not be very good English; moreover, there is an ambiguity in the original which does not exist in the translation. A doubt might arise in the mind of the learner which accents are meant, those which belong to the terms changed, or those which belong to the terms into which the change is made. In the translation the sense is precise, correct, and clear. Speaking of explanatory notes, the reviewer says, "in that given at page 95, doubtless by inadvertence, the parentheses, which ought to indicate the multiplication between the factors, are omitted." Parentheses in this case would be superfluous, the line separating the numerator from the denominator answering that purpose. In proof of this, examples might be quoted from writers of the first authority. Thus, page 82 of af- cd this very work, we have c b case in question, and which Cambridge, July, 1825. ae-bd perfectly similar to the is represented as faulty. To solve questions by the assistance of algebra Explanation of the words, equation, members, and terms - ib. ib. Resolution of Equations of the First Degree, having but one Rule for transposing any term from one member of an equation To disengage an unknown quantity from multipliers Of equations, the terms of which have divisors Rule for making the denominators in an equation to disappear To write a question in the form of an equation Examples ib. Methods for performing, as far as is possible, the operations indicated upon Quantities, that are represented by Letters Explanation of the terms, simple quantities, binomials, trinomials, Rule for performing addition - Rule for the reduction of algebraic quantities Subtraction of Algebraic Quantities Rule for performing subtraction b ib. Method of determining whether the root found is too small Whole numbers, except such as are perfect squares, admit of no assignable root, either among whole numbers or fractions What is meant by the term incommensurable or irrational How to denote by a radical sign, that a root is to be extracted The number of decimal figures in the square double the number Method of abridging, by division, the extraction of roots To approximate a root indefinitely, by means of vulgar fractions Most simple method of obtaining the approximate root of a frac- The square root of a quantity may have the signor General formula for resolving equations of the second degree, having only one unknown quantity Examples showing the properties of negative solutions In what cases problems of the second degree become absurd ib. 137 ib. Table of the first seven powers of numbers from 1 to 9 To obtain any power whatever of a simple quantity To extract the root of any power whatever of a simple quantity 138 Of fractional éxponents Of negative exponents Of the Formation of Powers of Compound Quantities - Manner of denoting these powers Form of the product of any number whatever of factors of the Method of deducing from this product the development of any Rule for the development of any power whatever of a binomial 149 General law for the number of roots of an equation, and the dis- tinction between arithmetical and algebraical determinations 166 Of Equations which may be resolved in the same manner as To determine their several roots |