cases to which Demonstration applies, are very few indeed compared with those which we are obliged to believe on Moral Evidence: we act upon the latter in nearly all the affairs of life: "it is frequently the only light afforded us to form our opinion of facts, and to regulate our conduct with respect to them. Without attending to Moral Evidence, we can neither act, nor cease to act; we cannot even subsist without acting upon it, since it cannot be demonstrated that our food will not poison instead of nourish us."

Thus much it was thought necessary to add on this subject, to caution the mathematical student never to attempt the extension of his demonstrative powers to objects beyond their proper sphere. Some have greatly erred in this particular, who, because they could demonstrate a few things, became so unaccountably enamoured with the seducing charms of demonstration, as scarcely to believe any thing on evidence short of mathematical proof: and from this absurd and dangerous scepticism, by an easy and natural transition, have at length terminated their earthly career in the gloomy and fearful character of confirmed atheists.

A farther Account of the Usefulnes of the Mathematics.

It is impossible by any description, to produce in the mind of one unacquainted with the subject, an adequate idea of the usefulness of mathematical learning; he alone who has made considerable progress in the acquisition, will be able in some sort to appreciate its value: it will however be necessary, for the information of the unlearned, to observe, that mathematical knowledge supplies most of the means by which the businesses and affairs of Society are carried on. To mathematical principles the skilful architect necessarily resorts for the means of contriving and executing his plans, and to them every building, from the cottage to the palace, owes its existence. By Mathematics, stately ships, which repel invasion from our coasts, assail with British thunder the fleets and maritime fortresses of our foes, or waft to the alternate shores the produce of every climate, are constructed. By it the mariner shapes his course through the wide and pathless ocean, and the intrepid soldier plans his operations for the honour and protection of our country, and the annoyance of our enemies: versed in mathematical knowledge, the miner contrives his subterraneous excavations, and performs

them with safety and success, procuring metals and minerals for our use, and fuel for our fires. By this the skilful Artist delineates with accuracy, faithful transcripts of Art and Nature; and the Musician extracts harmony from the animal, vegetable, and mineral Creation. Guided and assisted by mathematical reasoning, the ingenious Mechanic contrives and executes the various machines in use among us, and calculates their powers and effects; to mathematical principles almost every tool, or instrument we use is indebted for its existence or perfections, and by them, the extent and value of property of every description, are accurately computed and ascertained.

Hence it appears, that Mathematics, considered only with respect to its immediate application, is indispensably necessary to complete the philosopher, the mariner, the soldier, the mechanic, and the artist; but this excellent science, (although fundamental to them) is by no means confined to practical arts and necessary computations, it claims an important pre-eminence which renders it worthy the attention of all who aspire to the rank of rational beings. The Mathematics is a system of pure, strict, solid, and conclusive reasoning, calculated to strengthen, extend, and invigorate the powers of the mind *,

to

h The Rev. W. Jones very justly remarks, that," in philosophy, especially under the present state of it, the use of mathematical learning is unquestionable;" and Dr. Watts says,that "the moderns have found a thousand things by applying mathematics to natural philosophy, which the ancients were ignorant of." See Jones on the use of Math. Learning, p. 36; and Watts's Logic,

p. 80.

i Jones on mathematical learning.

k Dr. Watts observes, that "the greatest clearness of thought and force of reasoning abound in the mathematical sciences;" and he adds, that "if there were nothing valuable in them for the uses of human life, yet the very speculative parts are well worth our study; for by perpetual examples they teach us to conceive with clearness, and to conduct our ideas and propositions in a tram of dependence; to reason with strength and demonstration, and to distinguish between truth and falsehood." "The Mathematics" (says Mr. Locke) "is eminently serviceable to strengthen and improve the intellectual faculties, and fit them for every kind of speculation. Would you have a man reason well, you must use him to it betimes, exercise his mind in observing the connection of ideas, and following them in train; nothing does this better than Mathematics, which therefore, I think, should be taught all those who have time and opportunity, not so much to make them mathematicians, as to make them reasonable creatures.' "The ancients," says professor Duncan, "who so well understood the manner of forming the mind, always began with the Mathema. tics-here the understanding is by degrees habituated to truth, contracts in

"

guide and assist us in the investigation of truth; applicable to every enquiry where quantity is concerned, to which human reason is competent, and capable of conducting our researches with unerring certainty to the utmost limit to which the powers and faculties of the mind can extend.

From what has been said on the subject, the ingenious student will be able to form his own judgment; he will perceive that this science is of general utility, absolutely necessary to all who are engaged in any learned profession or art, useful in trade, and ought to be studied by the Physician, the Lawyer, the Divine, and by all who have sufficient opportunity.

But it must not hence be inferred, that every one is bound to

sensibly a certain fondness for it, and learns never to yield its assent to any proposition, but where the evidence is sufficient to produce full conviction. For this reason Plato has called mathematical demonstrations The Cathartics or Purgatives of the Soul, as being the proper means to cleanse it from error, and restore that natural exercise of its faculties." Lord Bacon represents the Mathematics as eminently adapted to correct the irregularities, and remedy the defects of the mind, equally beneficial to its faculties as athletic exercises are to those of the body, producing strength, vigour, and activity. Of the same opinion is Mr. Harris, who asserts that "every exercise of mind upon theorems of science, like generous and manly exercise of the body, tends to call forth and strengthen nature's original vigour. The nerves of reason are braced by the mere employ, and we become abler actors in the drama of life, whether our part be of the busier, or of the sedater kind." The Rev. W. Jones of Pluckley expresses similar sentiments in nearly the same words; and many other testimonies equally unexceptionable in behalf of mathematical learning might be added; but I shall conclude this long note with part of the Inaugural Oration of Dr. Barrow on his entering the Lucasian professorship of Mathematics at Cambridge in 1663. "The Mathematics," says he, "effectually exercise, not vainly delude, nor vexatiously torment studious minds with obscure subtilties, but plainly demonstrate every thing within their reach, draw certain conclusions, instruct by profitable rules, and unfold pleasant questions. These diciplines likewise inure and corroborate the mind to a constant diligence in study; they wholly deliver us from a credulous simplicity, most strongly fortify us against the vanity of scepticism, effectually restrain us from a rash presumption, most easily incline us to a due assent, and perfectly subject us to the government of right reason. While the mind is abstracted and elevated from sensible matter, distinctly views pure forms, conceives the beauty of ideas, and investigates the harmony of proportions, the manners themselves are insensibly corrected and improved, the affections composed and rectified, the fancy calmed and settled, and the understanding raised and excited to more divine contemplations."

aim at becoming a profound mathematician: our principal attention ought to be directed to the duties of the profession for which we are intended, or in which we are engaged, and to the acquirement of the parts of learning immediately connected with it.

Deep and difficult researches in the abstruse regions of science are indeed the business of a few, but they are not necessary to the greater portion of mankind; a general acquaintance with the sciences, provided the elementary branches be well understood, will be abundantly sufficient for the student, who does not intend to make this part of learning his chief study; it will enable him to understand the works on Philosophy, Astronomy, &c. which are usually read, and will be fully adequate to all the purposes of the general scholar.

A few Hints, which it is hoped will prove useful to the mathematical Student.

Our faculties, both mental and corporeal, are talents which our all-wise Creator, to accomplish his beneficent purposes, has committed to us in trust; and the cultivation of them is by no means optional; the duties we owe to God, our neighbour, and ourselves, require their utmost exertion, and strongly imply the necessity of their improvement, and that for the neglect, we shall one day stand accountable. The man possessing an improved understanding is certainly better qualified to benefit society than he who is ignorant; and if the obligation be equally binding on those who have learning, and those who possess the means, but want the inclination to obtain it, how culpable must he be who neglects or refuses instruction! What a pleasing advantage does the devout and serious mind enjoy, which is capable of tracing the Almighty in his magnificent works of creation! which, aided by the light and assistance of science, is enabled to contemplate the universal scale of being through its successive gradations from the minute to the stupendous, and at every step to discover convincing proofs of the power, the majesty, the wisdom, and the unchangeable goodness of God. This is equally the duty and the privilege of rational creatures, and although it ranks no higher than as a branch of natural religion, yet the sincere Christian will feel it of too great importance to be overlooked; it is perhaps the only view in

which he can contemplate sublunary things to advantage, and is fitted to supply one source of comfort, to cheer him under the difficulties and distresses of life.

In concluding this part of the subject, it will be proper to offer a few observations, whereby the student will be the better prepared to decide on the comparative value of the science here recommended.

Philosophy, or the knowledge of men and things, is the end to which learning aspires. The Mathematics (considered with regard to its practical application) is nearly the same to natural philosophy and the arts, as grammar is to language. Neither Mathematics nor grammar can teach its own use, nor be of any service except as introductory to subjects altogether distinct from, and superior to them.

Wisdom is the right use of knowledge, and ensures to us the attainment of the CHIEF GOOD; without wisdom all our proficiency in knowledge, as far as it concerns ourselves, will ultimately be of no advantage. Wisdom includes knowledge and rectitude of mind; knowledge is obtained by learning, especially by the mathematical sciences, which strengthen the mind, improve its faculties, and enlarge its powers in a wonderful manner; but rectitude of mind, without which knowledge is useless or pernicious, is not to be attained by science, or by any methods except those which the Holy Scriptures prescribe; this alone is calculated for the double purpose of stamping a just value on learning, and securing our eternal interests. Compare the longest life with eternity, and it will be found less than a drop of water when compared with the ocean: this should teach us in what proportion our anxiety for the present and the future ought to be regulated. But there is a happy art by which the interests of time and eternity may be both efficaciously pursued together, without obstructing each other. Let this be the student's chief aim, that, like a truly Christian Philosopher, he may "so pass through things temporal, as finally to lose not the things eternal.”

There is one more particular on which (as an instructor of youth, and feeling that I am accountable for the advice I give to a tribunal from whence there is no appeal) it is my duty to offer a few words. We highly value those by whose labours, inventions, or improvements in any useful branch of knowledge, have been