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And the great system instead of being presented as a whole, is divided into detached and separate parts, which are brought, one by one, in distinct individuality, before the mind. Expedients and processes are taught separately and expressly, -not brought up incidentally as the difficulties occur which they are intended to remove. In a word, the tendency now is to a return to a method based on its adaptedness to the limited and imperfect and undisciplined minds, whose wants are to be supplied rather than on the intrinsic and absolute nature of the principles of science. There can be no doubt that this return is a judicious one, and we are convinced that a work on the plan of Davies's or Peirce's, will be found altogether better adapted to use as a manual for class instruction than Euler's. We are not sure however that Davies does not go to the opposite extreme. For instance we think a class must be a very extraordinary class indeed to be carried to a state of mind of even tolerable satisfaction and repose, in respect to the whole subject of negative exponents, as it is disposed of on page 16, in a mere corrollary to what will appear to the pupil an accidental example. It is true the proposition=a", is rigidly demonstrated, but, most pupils, while they might understand, would not feel the force of the demonstration,-for it seems to us there is such a distinction. And then the very perplexing point, as it certainly appears to all beginners, and often to those that are not mere beginners, why such an expression as a- should be employed as equivalent to a., which forces itself upon the mind at least, seems worthy of some greater atttempt at explanation ; especially when we consider that the doctrine of negative exponents lies at the foundation of so important a part of the subsequent mathematical structure.
We do not mention this as an objection to the book so much as an example illustrating its character. It seems to be condensed, compact, rigid, in the highest degree, It is a text book, and the teacher must supply the commentary.
In respect to the theory of powers and exponents, we have never met with any satisfactory view of it. The different classes of exponents are generally treated entirely distinctly, i. e. so far as the foundation on which the notation is based : whereas they evidently belong to one and the same system, and ought to be brought into the same general
view. The source of the difficulty seems to be in the definition of the word power. Davies says, following the universal custom we believe, -" The power of a quantity is the product which results from multiplying the quantity by itself,” and an exponent, though not expressly defined to be such, is still represented to be the number that shows how many times the number is taken as a factor. This is applicable to all those powers whose indices are positive and plural, but we want a representation of the case which will include the whole class, for they all are evidently one in nature; such as
a>, a', a-2, at, a-, Only one of the above is the product of a quantity multiplied by itself, and of course only one is included under the definition. Now there certainly is a very clear and obvious analogy between these expressions, i. e. clear and obvious in its nature, however difficult it may be to clothe it in language. It links them together, forms of them one class, and enables us to combine the indices and operate upon them in every way, as numbers identical in their nature. This hidden analogy we ought to develop, and make it the foundation of the theory instead of giving a definition, as we always do now, which applies only to one of the above cases, and then empirically extends our reasonings and rules to the others.
In fact there is a confusion exactly analogous to this, farther back, in the ordinary attempts at defining multiplication. The true general idea of multiplication is not expressed, or even attempted to be expressed. Bailey says, as in fact most writers on arithmetic and algebra substantially do, “ Multiplication is merely a short way of performing addition, when the quantities to be added happen to be equal.” Peirce, Davies and Euler, as if instinctively shrinking from the difficulty, attempt no definition whatever. In respect, however, to the common definition, we may ask what addition is it that is performed in the case a x ļ, or a x 1, or a x-1? To this question it may perhaps be replied that the term multiplication is in strict propriety applied only to the case where the multiplier is positive and plural; and this may be a sufficient answer, though far from satisfactory to
In all the above cases the operations are strictly analogous, the term multiply is constantly used by all writers in respect to each of them, and we cannot but think that the perfection of science requires that this common analogy
should be made the foundation of the definition and the theory. It ought however to be stated, distinctly that the authors of these works are not in fault at all, in respect to these points. It is the present state of the science that we are criticising, and not the success of their attempts to ex. hibit it. In respect however to the theory of powers and the analogy of positive, negative, integral and fractional exponents, we had some further suggestions to make, but must postpone them to some future occasion.
ART. III.- ACADEMIES IN NEW ENGLAND.
ANOTHER defect in the constitution of the Academies of New England is the want of funds. From this want arises that dependence on popular favor, which, if it require these institutions to keep pace with the improvement, real or fanciful, of the age, also and for that very reason, forbids the establishment and enforcement of any thorough plan of discipline, or any permanent system of instruction. The resources of these institutions are in most instances too meagre to support an adequately qualified teacher, and those who are not willing to be subject to the caprice of patronage, seek a surer competence in a steadier occupation. The same difficulty substantially is felt in nearly every school and college in the country. In many of the latter the Professors are straitened for the means of subsistence. The consequence is obvious. The dependent cannot well be manly, and in a cause which most of all ought to be kept aloof from the changing influences of party and of whim, the seeking of favor introduces servility, and the clamors of the ignorant are more regarded than the decisions of the learned. We have ample proof of the utility of permanent endowments in the instance of the few academies among us which are thus provided. Were it not invidious we could select instances thoroughly in point.
We have said that academies are like to be continued in New England for many years. Many of the larger towns in Massachusetts and probably in other States, have recently established High Schools, which will draw off many students who have been accustomed to resort to academies. This, however, will effect no material change in the system of instruction hitherto practised, and certainly will not, we think, diminish the number of academies. If this scheme of instruction, which has been a favorite one in New England, is to be continued, the establishment of it on a foundation of stable utility is a duty which the friends of it will hardly attempt to evade.
We have no full statistical account of the incorporated academies in New England. The following statement will be found, we think, not far from accurate. In Maine are 40, more than half of which are endowed with 11,500 acres of land. In New Hampshire, about 40, several of which have large funds. In Vermont, between 20 and 30, slightly or not at all endowed. In Massachusetts, about 60, 23 of which have received from the State a tract of land in Maine, six miles square. In Connecticut, about 30. C. C.
We annex to the foregoing article, a list of academies in New Hampshire and Massachusetts, taken from the American Quarterly Register, and amended according to our best knowledge. The list was made several years ago and some changes may have been made of which we are not aware.
New Hampshire. The Adams female, Derry, was incorporated 1823; Alstead, 1516; Atkinson, 1791 ; Boscawen, 1828; Brackett, Greenland, 1824; Chesterfield, 1790; Effingham, 1819; Francestown, 1819; Franklin, Dover, 1803; Gilford, 1820; Gilmanton, 1794 ; Hampton, 1810; Haverhill
, 1794 ; Hillsborough, 1821 ; Holmes, Plymouth, 1808; Hopkinton, 1827; Kimball Un. Plainfield, 1813; Lancaster, 1808; New Hampton, 1821 ; New Ipswich, 1789; Newport, 1819; Pembroke, 1818; Phillips, Exeter, 1781 ; Pinkerton, Derry, 1814; Portsmouth, 1808 ; Salisbury, 1808; Rochester, 1827; Wakefield, 1827 ; Walpole, 1831; Wolfeboro' and Tuftonboro' 1820; Woodman, Sanbornton, 1820.
Phillips' Exeter academy was founded at Exeter, by the Hon. John Phillips, LL. D. It is one of the best endowed institutions of the kind in the United States. It has a library of 600 volumes and a valuable philosophical apparatus. The building is an edifice 76 by 36 feet, two stories high, with two wings, 34 by 28 feet, one story high. The number of students is 75. The Adams female academy in Derry, has a fund of $4,000. It has a good chemical and ph:10sophical apparatus. All the branches of an English education are taught with, the Latin and French languages. The Gilmanton academy has funds-6,000 dollars at interest, and 7,000 acres of land in Coos county. The Kimball Union academy has 40,000 dollars in funds, the donation of Hon. Daniel Kimball. The income is devo ed principally to aid pious and indigent young men in preparing for the Christian ministry. The Pinkerton academy was founded by Major John Pinkerton. Funds, 15,000 dollars, besides real estate.
Massachusetts. The academy at Williamstown was incorporated in 1828 ; the Pittsfield female academy in 1807; the Stockbridge academy in 1828; the Lenox academy, incorporated in 1803, has prepared a large number of individuals for college, and is a very useful institution ; the average number of scholars, 60 or 70; the Northfield academy has 107 students and the annual expense for instruction, &c. is $800. At Greenfield is the “ Fellenberg institution" under the instruction of Mr James H. Coffin ; the students are essentially aided by provisions for manual labor. Deerfield academy is one of the oldest in the State, and was incorporated in 1797 ; it has a valuable chemical and philosophical apparatus. Amherst academy was incorporated in 1816; the number of scholars is from 90 to 120, all males ; a class of 20 or 30 are fitted for college each year; it has been ever since its establishment one of the principal academies in the State. At Hadley is Hopkins academy, incorporated in 1816; the income from the funds amounts to about $100 per annum. At Southampton, eight miles south of Northampton, is the “Sheldon academy," incorporated in 1829. Westfield academy was incorporated in 1793; the academy is provided with a chemical and philosophical apparatus ; lectures are given on a variety of subjects; the academy has a fund, the income of which is applied to the payment of teachers in part. At Springfield $600 is paid annually for the support of a high school. At Wilbraham, is the Wesleyan seminary, incorporated in 1824, and a flourishing institution, em