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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-?, 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 us. 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 exhibit 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, 1816; 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 chemal and philosophical 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 Wilbrahan, is the Wesleyan seminary, incorporated in 1824, and a flourishing institution, embracing males and females, and a various course of study. At Monson is a very flourishing institution; the half township of land given to this academy was sold for $5,000; attached to the institution is a general fund of $6,000, a premium fund of $500, and a charity fund of $6,500, making in all $13,000; the charity fund is designed to aid young men in preparing for the ministry ; facilities are enjoyed at this academy for manual labor; board is very reasonable. At Leicester is one of the oldest academies in the State, incorporated in 1784 ; the funds amount to $19,000; average number of scholars, 60 or 70; a new building has, within a few years, been erected for the use of this academy. At Dudley is Nichols academy, incorporated in 1819. At Milford is an academy, incorporated in 1828, which has about 35 scholars each quarter. At Westminster is an academy, incorporated in 1833, which has 25 scholars, about one half from the neighboring towns. The academy at New Salem was incorporated in 1795 ; the Gates in Marlboro' in 1830, funds, $2,000; the Framingham in 1799, funds $7,000 ; the Billerica in 1820; the Groton in 1793; the Lancaster in 1828; the Lexington in 1822; the Westford in 1793; the Middlesex female at Concord in 1806; the Haverhill in 1823; Central village academy in Dracut, in 1833; the Bradford academy in the west parish of Bradford, in 1804. The Dummer academy at Newbury, incorporated in 1782, has large funds, given by the gentleman whose name it bears. The Newburyport academy, incorporated in 1807. The Ipswich female seminary was incorporated in 1828. It is the leading object of the seminary to prepare young ladies of mature minds for active usefulness, especially to become teachers ; none are received under the age of 14 years. The winter term commences on the last Wednesday in October, and continues 25 weeks, including a vacation of one week. The summer term commences the last Wednesday in May, and continues 16 weeks. At Topsfield is an academy incorporated in 1828 ; Marblehead in 1792 ; at Lynn incorporated in 1805; at North Andover, the Franklin academy, incorporated in 1803; at East Bradford, the Merrimac, incorporated in 1822. Phillips, at Andover, south parish, was incorporated in 1780, and has two departments, classical and English. The institution is provided with a respectable building and with a library of several hundred volumes ; the