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only, is called arithmetic; and that which is concerned about measure in general, whether length, breadth, motion, force, &c., is called geometry.
As to the usefulness of arithmetic, it is well known that no business, commerce, trade, or employment whatsoever, even from the merchant to ihe shopkeeper, &c., can be managed and carried on without the assistance of numbers; for by these the trader computes the value of all sorts of goods that he dealeth in, does his business with ease and certainty, and informs himself how. matters stand at any time with respect to men, money, and merchandise, to profit and loss, whether he goes forward or backward, grows richer or poorer. Neither is this science only useful to the merchant, but is reckoned the primum mobile (or first mover) of all mundane affairs in general, and is useful for all sorts and degrees of men, from the highest to the lowest.
As to the usefulness of geometry, it is as certain that no curious art or mechanic work can either be invented, improved, or performed without its assisting principles.
It is owing to this that astronomers are put into a way of making their observations, coming at the knowledge of the extent of the heavens, the duration of time, the motions, magnitudes, and distances of the heavenly bodies, their situations, positions, risings, settings, aspects, and eclipses; also the measure of seasons, of years, and of ages.
It is by the assistance of this science that geographers present to our view at once the magnitude and form of the whole earth, the vast extent of the seas, the divisions of empires, kingdoms, and provinces.
It is by the help of geometry the ingenious mariner is instructed how to guide a ship through the vast ocean, from one part of the earth to another, the nearest and safest way, and in the shortest time.
By help of this science the architects take their just measures for the structure of buildings, as private houses, churches, palaces, ships, fortifications, &c.
By its help engineers conduct all their works, take the situation and plan of towns, forts, and castles, measure their distances from one another, and carry their measures into places that are only accessible to the eye.
From hence also is deduced that admirable art of drawing sundials on any place, howsover situate, and for any part of the world, to point out the exact time of the day, the sun's declination, altitude, amplitude, azimuth, and other astronomical matters.
By geometry the surveyor is directed how to draw a map of any country, to divide his lands, and to lay down and plot any piece of ground, and thereby discover the area in acres, rods, and perches; the gauger is instructed how to find the capacities or solid contents of all kinds of vessels, in barrels, gallons, bushels, &c.; and the measurer is furnished with rules for finding the areas and contents of superfices and solids, and casting up all manner of workmanship. All these, and many more useful arts, too many to be enumerated here, wholly depend upon the aforesaid sciences, namely, arithmetic and geometry.
This science is descended from the infancy of the world, the inventors of which were the first propagators of human kind, as Adam, Noah, Abraham, Moses, and divers others.
There has not been any science so much esteemed and honoured as this of the mathematics, nor with so much industry and vigilance become the care of great men, and laboured in by the potentates of the world, namely, emperors, kings, prinMathematical demonstrations are a logic of as
much or more use than that commonly learned at schools, serving to a just formation of the mind, enlarging its capacity, and strengthening it so as to render the same capable of exact reasoning, and discerning truth from falsehood in all occurrences, even subjects not mathematical. For which reason it is said the Egyptians, Persians, and Lacedæmonians seldom elected any new kings but such as had some knowledge in the mathematics; imagining those who had not men of imperfect judgments, and unfit to rule and govern.
Though Plato's censure, that those who did not understand the 117th proposition of the 13th book of Euclid's Elements ought not to be ranked among rational creatures, was unreasonable and unjust, yet to give a man the character of universal learning, who is destitute of a competent knowledge in the mathematics, is no less so.
The usefulness of some particular parts of the mathematics, in the common affairs of human life, has rendered some knowledge of them very necessary to a great part of mankind, and very convenient to all the rest, that are any way conversant beyond the limits of their own particular callings.
Those whom necessity has obliged to get their bread by manual industry, where some degree of art is required to go along with it, and who have had some insight into these studies, have very often found advantages from them sufficient to reward the pains they were at in acquiring them. And whatever may have been imputed to some other studies, under the notion of insignificance and loss of time, yet these, I believe, never caused repentance in any, except it was for their remissness in the prosecution of them.
Philosophers do generally affirm that human knowledge to be most excellent which is conversant among the most excellent things. What science, then, can there be more noble, more excel
lent, more useful for men, more admirably high and demonstrative, than this of the mathematics ?
I shall conclude with what Plato says, in the seventh book of his Republic, with regard to the excellence and usefulness of geometry, being to this purpose:
“ Dear friend-You see, then, that mathematics are necessary, because, by the exactness of the method, we get a habit of using our minds to the best advantage. And it is remarkable that, all men being capable by nature to reason and understand the sciences, the less acute, by studying this, though useless to them in every other respect, will gain this advantage, that their minds will be improved in reasoning aright; for no study employs it more, nor makes it susceptible of attention so much; and those who we find have a mind worth cultivating, ought to apply themselves to this study."
THE ART OF PROCURING PLEASANT
Inscribed to Miss
being written at her request.
As a great part of our life is spent in sleep, during which we have sometimes pleasant and sometimes painful dreams, it becomes of some consequence to obtain the one kind and avoid the other; for, whether real or imaginary, pain is pain and pleasure is pleasure. If we can sleep without dreaming, it is well that painful dreams are avoided. If, while we sleep, we can have any pleasing dreams, it is, as the French say, autant de gagné, so much added to the pleasure of life.
To this end it is, in the first place, necessary to
be careful in preserving health, by due exercise and great temperance; for in sickness the imagination is disturbed, and disagreeable, sometimes terrible, ideas are apt to present themselves. Exercise should precede meals, not immediately follow them; the first promotes, the latter, unless moderate, obstruct's digestion. If, after exercise, we feed sparingly, the digestion will be easy and good, the body lightsome, the temper cheerful, and all the animal functions performed agreeably. Sleep, when it follows, will be natural and undisturbed; while indolence, with full feeding, occasions nightmares and horrors inexpressible; we fall from precipices, are assaulted by wild beasts, murderers, and demons, and experience every variety of distress. Observe, however, that the quantities of food and exercise are relative things; those who move much may, and, indeed, ought to, eat more; those who use little exercise should eat little. In general, mankind, since the improvement of cookery, eat about twice as much as nature requires. Suppers are not bad if we have not dined; but restless nights naturally follow hearty suppers after full dinners. Indeed, as there is a difference in constitutions, some rest well after these meals; it costs them only a frightful dream and an apoplexy, after which they sleep till doomsday. Nothing is more common in the newspapers than instances of people who, after eating a hearty supper, are found dead abed in the morning.
Another means of preserving health, to be attended to, is the having a constant supply of fresh air in your bedchamber. It has been a great mistake, the sleeping in rooms exactly elosed, and in beds surrounded by curtains. No outward air that may come in to you is so unwholesome as the unchanged air, often breathed, of a close chamber, As boiling water does not grow hotter by longer boiling, if the particles that receive greater heat