AN ESSAY

ON THE

USEFULNESS OF MATHEMATICAL LEARNING,

IN A LETTER FROM A GENTLEMAN IN THE CITY TO HIS FRIEND IN OXFORD'.

I AM glad to hear from you that the study of the mathematics is promoted and encouraged among the youth of your University. The great influence which these sciences have on philosophy and all useful learning, as well as the concerns of the public, may sufficiently recommend them to your choice and consideration; and the particular advantages which you of that place enjoy, give us just reason to expect from you a suitable improvement in them. I have here sent you some short reflections upon the usefulness of mathematical learning, which may serve as an argument to incite you to a closer and more vigorous pursuit of it.

In all ages and countries where learning hath prevailed, the mathematical sciences have been looked upon as the most considerable branch of it. The very name Má@noris implies no less; by which they were called either for their excellency, or because of all the sciences they were first taught, or because they were judged to comprehend πάντα τὰ Μαθήματα, And amongst those that are commonly reckoned to be the seven liberal arts, four are mathematical, to wit, arithmetic, music, geometry, and astronomy.

But notwithstanding their excellency and reputation, they have not been taught nor studied so universally as some of the rest; which I take to have proceeded from the following causes : the aversion of the greatest part of mankind to serious attention and close arguing; their not comprehending sufficiently the necessity or great usefulness of these in other parts of learning; 1 See pages, 24-26.

an opinion that this study requires a particular genius and turn of head, which few are so happy as to be born with; and the want of public encouragement, and able masters. For these, and perhaps some other reasons. this study hath been generally neglected, and regarded only by some few persons whose happy genius and curiosity have prompted them to it, or who have been forced upon it by its immediate subserviency to some particular art or office.

Therefore I think I cannot do better service to learning, youth, and the nation in general, than by shewing that the mathematics of all parts of human knowledge, for the improvement of the mind, for their subserviency to other arts, and their usefulness to the commonwealth, deserve most to be encouraged. I know a discourse of this nature will be offensive to some, who, while they are ignorant of mathematics, yet think themselves masters of all valuable learning; but their displeasure must not deter me from delivering an useful truth.

The advantages which accrue to the mind by mathematical studies consist chiefly in these things: first, in accustoming it to attention; secondly, in giving it a habit of close and demonstrative reasoning; thirdly, in freeing it from prejudice, credulity, and superstition. First, the mathematics make the mind attentive to the objects which it considers. This they do by entertaining it with a great variety of truths which are delightful and evident, but not obvious. Truth is the same thing to the understanding, as music to the ear, and beauty to the eye. The pursuit of it does really as much gratify a natural faculty implanted in us by our wise Creator, as the pleasing of our senses; only in the former case, as the object and faculty are more spiritual, the delight is the more pure, free from the regret, turpitude, lassitude and intemperance that commonly attend sensual pleasures. The most part of other sciences consisting only of probable reasonings, the mind has not where to fix; and wanting sufficient principles to pursue its searches upon, gives them over as impossible. Again, as in mathematical investigations truth may be found, so it is not always obvious this spurs the mind, and makes it diligent and attentive. In Geometria (says Quintilian, 1. i. c. 10) partem fatentur esse utilem teneris aetatibus: agitari namque animos, atque acui ingenia, et celeritatem percipiendi venire inde concedunt. And Plato (in Repub. lib. vii.) observes, that the youth who

are furnished with mathematical knowledge are prompt and quick at all other sciences, εἰς πάντα τὰ Μαθήματα ὀξεῖς φαίνονται. Therefore he calls it, κατὰ παιδείαν ὁδόν. And indeed youth is generally so much more delighted with mathematical studies than with the unpleasant tasks that are sometimes imposed upon them, that I have known some reclaimed by them from idleness and neglect of learning, and acquire in time a habit of thinking, diligence, and attention; qualities which we ought to study by all means to beget in their desultory and roving minds.

The second advantage which the mind reaps from mathematical knowledge, is a habit of clear, demonstrative, and methodical reasoning. We are contrived by nature to learn by imitation more than by precept; and I believe in that respect reasoning is much like other inferior arts (as dancing, singing, &c.) acquired by practice. By accustoming ourselves to reason closely about quantity, we acquire a habit of doing so in other things. It is surprising to see what superficial inconsequential reasonings satisfy the most part of mankind. A piece of wit, a jest, a simile, or a quotation of an author, passes for a mighty argument with such things as these are the most part of authors stuffed, and from these weighty premises they infer their conclusions. This weakness and effeminacy of mankind in being persuaded where they are delighted, have made them the sport of orators, poets, and men of wit. Those lumina orationis are indeed very good diversion for the fancy, but are not the proper business of the understanding; and where a man pretends to write on abstract subjects in a scientifical method, he ought not to debauch in them. Logical precepts are more useful, nay, they are absolutely necessary for a rule of formal arguing in public disputations, and confounding an obstinate and perverse adversary, and exposing him to the audience, or readers. But in the search of truth, an imitation of the method of the geometers will carry a man further than all the dialectical rules. Their analysis is the proper model we ought to form ourselves upon, and imitate in the regular disposition and progress of our enquiries; and even he who is ignorant of the nature of mathematical analysis uses a method somewhat analogous to it. The composition of the geometers, or their method of demonstrating truths already found out, viz. by definitions of words agreed upon, by self-evident truths, and

propositions that have been already demonstrated, is practicable in other subjects, though not to the same perfection, the natural want of evidence in the things themselves not allowing it; but it is imitable to a considerable degree. I dare appeal to some writings of our own age and nation, the authors of which have been mathematically inclined. I shall add no more on this head, but that one who is accustomed to the methodical systems of truths, which the geometers have reared up in the several branches of those sciences which they have cultivated, will hardly bear with the confusion and disorder of other sciences, but endeavour as far as he can to reform them.

Thirdly, mathematical knowledge adds a manly vigour to the mind, frees it from prejudice, credulity, and superstition. This it does two ways: first, by accustoming us to examine, and not to take things upon trust; secondly, by giving us a clear and extensive knowledge of the system of the world; which, as it creates in us the most profound reverence of the almighty and wise Creator, so it frees us from the mean and narrow thoughts which ignorance and superstition are apt to beget. How great an enemy mathematics are to superstition appears from this, that in those countries, where Romish priests exercise their barbarous tyranny over the minds of men, astronomers, who are fully persuaded of the motion of the earth, dare not speak out; but though the Inquisition may extort a recantation, the Pope and a General Council too will not find themselves able to persuade to the contrary opinion. Perhaps this may have given occasion to a calumnious suggestion, as if mathematics were an enemy to religion, which is a scandal thrown both on the one and the other; for truth can never be an enemy to true religion, which appears always to the best advantage when it is most examined :

Si propius stes

Te capiet magis.

On the contrary, the mathematics are friends to religion; inasmuch as they charm the passions, restrain the impetuosity of imagination, and purge the mind from error and prejudice. Vice is error, confusion and false reasoning; and all truth is more or less opposite to it. Besides, mathematical studies may serve for a pleasant entertainment for those hours which young men are apt to throw away upon their vices; the delightfulness

of them being such, as to make solitude not only easy, but desirable.

What I have said may serve to recommend mathematics for acquiring a vigorous constitution of mind; for which purpose they are as useful, as exercise is for procuring health and strength to the body. I proceed now to shew their vast extent and usefulness in other parts of knowledge. And here it might suffice to tell you, that mathematics is the science of quantity, or the art of reasoning about things that are capable of more and less, and that the most part of the objects of our knowledge are such as matter, space, number, time, motion, gravity, &c. We have but imperfect ideas of things without quantity, and as imperfect a one of quantity itself without the help of mathematics. All the visible works of God Almighty are made in number, weight, and measure; therefore to consider them we ought to understand arithmetic, geometry, and statics; and the greater advances we make in those arts, the more capable we are of considering such things as are the ordinary objects of our conceptions. But this will farther appear from particulars.

And first, if we consider to what perfection we now know the courses, periods, order, distances, and proportions of the several great bodies of the universe, at least such as fall within our view, we shall have cause to admire the sagacity and industry of the mathematicians, and the power of numbers and geometry well applied. Let us cast our eyes backward, and consider astronomy in its infancy; or rather let us suppose it still to begin for instance, a colony of rude country people, transplanted into an island remote from the commerce of all mankind, without so much as the knowledge of the calendar, and the periods of the seasons, without instruments to make observations, or any the least notion of observations or instruments. When is it we could expect any of their posterity should arrive at the art of predicting an eclipse? Not only so, but the art of reckoning all eclipses that are past or to come, for any number of years? When is it we could suppose that one of those islanders transported to any other place of the earth, should be able by the inspection of the heavens to find how much he were south or north, east or west of his own island, and to conduct his ship back thither? For my part, though I know this may be, and is daily done, by what is