of the hill following after as it were, and urging us on. So secundus ventus . . . secundo Alumine, &c.' · In the case of the wind and the stream the origin of the phrase is evident; but we confess we cannot readily form a notion of a hill running after us, at least without thinking of an avalanche in Switzerland, or some of the bogs in Ireland. Mr. Anthon will perhaps ask us for our own explanation; but we must decline giving any until we find authority for the phrase, which we do not at the present moment recollect ever to have met with. Though Mr. Anthon's edition is, without comparison, superior to that by Mr. Trollope, still they have some defects in common. We have already pointed out Mr. Trollope's regard for certain grammatical terms borrowed from the Greek tongue: the following quotations will show that Mr. Anthon has a leaning the same way: : Jug. 3. 41. Coepere nobilitas dignitatem populus libertatem in lubidinem vertere. Note: an elegant zeugma operates in lubidinem. Note: a C. 49. Ut quemque pecunia aut honore extulerat. zeugma operates in extulerat, by which it assumes a separate meaning with both pecunia and honore. "As he had gifted (?) any one with a present of money, or distinguished him by promotion.' The zeugma, however, may be avoided, if extulerat be rendered" he had distinguished." But this is less elegant.' See also c. 29. uti acciperet, and c. 42. multos mortalis. Again: C. 14. Capti ab Jugurtha, pars in crucem acti, pars bestiis objecti. Note: capti agreeing in gender with adfines, amici, &c. to which also acti and objecti refer by synesis.' Similar notes are given on c. 16. pars illa, qui, &c., and c. 95. magno equitatu-quos, &c. A reference from one to another of these passages would have been, we think, a more useful illustration for the American student, than the apparent solution of a difficulty by calling in the assistance of certain Greek words. There are other terms of the same nature we might point out, such as metonymy, archaisms, &c. The other point, wherein our two editors agree, is the practice of explaining grammatical difficulties, by understanding, or, as Mr. Trollope would express it, by subauding certain convenient words. Every page of the notes would furnish examples :— Jug. c. 48. Humi arido, understand solo to govern humi in the genitive.-c. 84. Plerosque militiae cognitos. Militiae, sc. in tempore' Or, again : Jug. c. 30. Apud plebem gravis invidia; patres probarentne flagitium, an decretum subverterent, parum constabat.' Upon this we find the following note : 'Cortius places a comma after patres, which then becomes either the nominative absolute, or else the accusative governed by quod ad understood. The punctuation we have adopted is decidedly preferable. "It was uncertain whether the Senate would approve," &c.' The explanation, by supplying the words quod ad, can scarcely be called an explanation at all, seeing that quod ad patres is a phrase wholly inadmissible. (In c. 92. is another instance, where Mr. Anthon avails himself of the same explanation.) All, perhaps, that Cortius meant by the unfortunate comma, was this, that the reader should throw an emphasis upon the word patres, as opposed to plebem; and, after an emphatic word, a pause is always made in practice, no matter whether it be noted in our system of punctuation or not. We object also to Mr. Anthon's translation, for the very reason that this emphasis is neglected. Had the order of the English corresponded more closely to that of the original, this defect would have been avoided; and this leads us generally to complain of the practice in our schools of transposing the words of a Latin author in translating, so as to reduce them to what is conceived to be the natural order of the English tongue. Some of our school books go even so far as to lay down certain rules for this re-arrangement. A boy is first to hook up a nominative; then, we believe, he is to bait for a verb; and so on. Now we feel assured that if the words were translated somewhat more in the order in which they present themselves, the meaning of a passage would more readily be found, and the translation would be more likely to retain the spirit of the original. So far we have given a somewhat unfavourable view of Mr. Anthon's Sallust. It remains to lay before the reader extracts from such of his notes (and they constitute a considerable portion) as deserve commendation. Those on geography are, almost without exception, far superior to any we meet with in our English editions. 6 The following is the note on Zama, Jug. c. 56 :- Zama, a city of Numidia, five days' journey west of Carthage, according to Polybius (xv. 5.). Near this place Scipio, subsequently surnamed Africanus Major, obtained a decisive victory over the Carthaginian forces under the command of Hannibal. Strabo and Hirtius speak of it as the royal residence of Juba. It was levelled to the ground by the Romans after the death of Juba, but rebuilt in the reign of Hadrian, and by his orders. No traces of it remain at the present day, &c. &c.' We give another note of the same nature: Jug. c. 21. Cirtam. 'Cirta, now Constantina, a city of Numidia on the river Ampsagas, at a considerable distance from the coast. It appears to have been originally the only important city of the more inland parts of Numidia, and hence, probably, its name from the Punic kartha, "a city." It was the royal residence of the kings of Numidia, of whom Micipsa, according to Strabo, did the most to enlarge and improve it. Compare the words of the geographer (Strab. 17. vol. vi. p. 669.) It was afterwards called Sittianorum Colonia, from P. Sittius Nucerinus, who greatly assisted Cæsar in the African war, and was rewarded for his services with the city and district. Compare note on Catiline, c. 21.' In the notes on luxu (dative), c. 6, and die (genitive), c. 21, we have the best kind of illustration in a collection of passages where the same forms occur. There is an excellent note of the same character on c. 16, in reference to the words fama, fide, the length of which alone prevents us from quoting it. Moreover, Mr. Anthon takes considerable pains in referring his reader to the other authorities for the different historical facts given in the text of his author. Notes of this kind have the twofold advantage of brevity and utility. On the whole, we find it extremely difficult to come to a conclusion as to the merits of Mr. Anthon's edition. There is much that is good in it; there is also much that might be improved or totally erased. With this qualified opinion, we must leave the book in the hands of our readers. ELEMENTS OF ALGEBRA. The Elements of Algebra, designed for the Use of Eton School; by the Rev. John Bayley, M.A., late Fellow and Mathematical Lecturer at Emanuel College, Cambridge.London: Whitaker, Treacher, and Co. THIS is a book for the use of Eton, well printed, and imposing in its appearance. We can now see how algebra is taught in one of our largest public schools. It opens with a definition of algebra, and an explanation of algebraic characters,' in which the most material omission is, what is meant by the letters that are put down. It begins, The number prefixed to an algebraic quantity is called its coefficient; but what an algebraic quantity is, this deponent sayeth not. It proceeds to the symbol-7 ax, which is called a quantity; while, just before, it is said that when one quantity is to be subtracted from another, it is preceded by this sign (-).' This mystification goes on throughout, and its frequency in elementary works is thought to palliate its absurdity. The work is divided into two parts; the first consisting entirely of rules and examples, the second of the same processes repeated more rationally. Strange to say, there is nothing on simple equations in the first part, though qua dratic equations are introduced. Ratio is said to be the relation, in point of magnitude, which one quantity bears to another of the same kind. For instance! the ratio of (a) to (b), expressed thus (a: b), denotes the magnitude of (a) with respect to (b); and when two ratios are equal to one another, the four quantities composing them are said to be proportionals.' Here is the old definition of Euclid, which is allowed to mean nothing; but it is not here, as in Euclid, that the unmeaning phrase is followed by a test of equality of ratios which supersedes the preceding definition. The moment after we come to this-It appears from the definition that We confess we cannot see this. α C b d The extension of the law of exponents to fractional powers is assumed, and not proved. The imaginary expressions a, &c. are introduced without the slightest notice of their nature, except what is contained in the name imaginary. The author seems to think that it is justifiable to convert a theorem into a definition, for we find, if a room be 8 yards long and 5 yards broad, the floor is said to contain 8 x 5 (or 40) square yards.' Now, the only conventional part of this is the saying square yard' instead of 'square whose side is a yard;' the remainder is a demonstrable theorem. But as if to make up for sometimes admitting as a definition what ought to be proved as a theorem, we find a contrary course pursued, and that which is conventional stated as a result of demonstration. For example :1 x3 =x-1; but -; therefore is the same X.4 ز 2.3 1 as a1. In like manner it may be shown that as x--2,' X We find in Logarithms the following:- If we assume a certain quantity, a, with a variable index, x, it is evident that, by taking every possible value of x, the quantity a* ma represent all numbers whatever.' In the first place, this is not correct; in the second place, if it were, it is not evident. Can the author tell us what value must be given to x to make 109? He will answer that he can take x so as to bring 10 as near to 9 as we please. That is true, but it is not what he asserted. There is a chapter on arithmetical notation at the end of the first part, which is better than the rest; but we do not see the use, in so small a treatise, of establishing a formula for perfect numbers. The second part contains proofs of many of the rules in the first, and commences, very rationally, with simple equations, this being in reality the most simple part of algebra. The whole is easy and correct until we come to the chapter on involution. Here the binomial theorem is proved; and, though we do not object to the execution of this part, we cannot see how pupils trained after the preceding methods can understand it. After several chapters, we come again to the subject of logarithms, where we find an oversight of this nature. The development of (1+b), arranged in powers of x, is asserted to be X2 + b −&c.) + 2.2 (b − b3 + b1 − &c.) +Raa + &c. 2 3 2 1.2 The coefficient of is totally wrong; it should be b—b3 + 11 12 as (l 1.2 -&c., being, in fact, as is afterwards proved, the same b2 + &c.) Surely the author should have 2 3 observed the remarkable form of (b-b+b &c.), as he evidently means the law established in the two last terms to continue; and, had he really satisfied himself of its truth, he should have caught at the immediate consequence, which is (Log. b) b − ; a result hitherto unsuspected by mathematicians, and which would have established his fame as an analyst to the end of time. The author of this treatise must excuse our speaking plainly about his book, of which we can neither approve the plan nor the execution, except that, as to the latter, the rules given are free from ambiguity. He is a man of talent, meant for better things than combining (we cannot add arranging) the numerous absurdities prevalent in our algebraical works. He owes it to his situation not to let |