THE SCIENCE AND ART OF ARITHMETIC· For the Use of Schools. PART I. INTEGRAL. BY A. SONNENSCHEIN AND H. A. NESBITT, M.A., UNIV. COLL., LONDON. "The mills of God grind slowly, but they grind exceeding small." BIBLIOTHECA NEW STEREOTYPED EDITION 1881 GODLEIANA LONDON: W. SWAN SONNENSCHEIN & ALLEN; WHITTAKER & Co.; SIMPKIN & Co.; HAMILTON & Co.; KENT & Co. LIVERPOOL: PHILIP, SON & NEPHEW. [All Rights reserved.] 181. f. 60 TO MISS PIPE, OF LALEHAM, CLAPHAM PARK, This Work is inscribed, IN TOKEN OF THEIR RESPECT AND GRATITUDE, BY THE AUTHORS PREFACE TO THE FIRST EDITION. THERE is no need now to insist on a rational study of Arithmetic. It is admitted on all sides that no subject is so well fitted for the early training of the reasoning powers, and principally because the student is enabled, without apparatus of any kind, steadily to test all his a priori conclusions by the light of experience. In History, Physics, and even in Language, the student must have premisses supplied him; but his Mathematical studies can all be "evolved from his inner consciousness." Ever since the pernicious plan of teaching by mere rote and rulo of thumb was abandoned, the teaching of Elementary Mathematics has steadily risen towards higher levels, and we may perhaps be allowed to note down some of the most remarkable stages. A certain school of teachers very early felt the necessity of enlisting the child's reason on their side; but the means they adopted were not always wise or even honest. In a modest little work on Vulgar Fractions, which is otherwise very meritorious, we find the following "proof” of the formula ==========÷, &c. If of both α ma b mb 5 8 89 terms be multiplied by 3, we obtain &; by 4, §; and so on. If, on the contrary, of 10, both terms be divided by 5, we obtain, , &c.; hence (!), &c. &c. We can imagine that inexperienced children would readily give their assent to such a proof; but the teacher ought to have known, 1st, that,,, &c., are only integers very thinly disguised as fractions, and that it does not follow that what is true of integers is |