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ther as the squares of their corresponding sides, the areas of the circles will also be proportional to the squares of their radii.
Thus, as the square of the hypothenuse is equal to the squares of the two sides, a circle, of which this hypothenuse is the radius, will be equal to two circles having for their radii the two other sides. The knowledge of this enables you to construct a basin of water as large as two other basins together.
The circle can be doubled exactly, though it cannot be exactly squared.
When accustomed thus to feel the advantages of geometrical truths, the pupil reads in some elements of this science, that if a straight line, called a tangent, be drawn touching a circle in one point, another straight line can never be made to pass between this circle and this line. This is evident enough, and was scarcely worth the trouble of saying. But it is added, that an infinite number of curve lines may
be made to pass through this point of contact. This surprises him; and it would surprise older persons: he is tempted to believe that matter is penetrable. The books tell him that this is not matter, that these are lines without breadth. But if they are without breadth, these metaphysical straight lines will pass one upon another for ever without touching anything. If they have breadth no curve can pass. The child no longer knows where he is; he finds himself transported into a new world, which has nothing in common with our own,
How shall he believe, that what is manifestly impossible in nature, is true?
I well conceive, he will say to a master of the transcendental geometry, that all your circles will meet in C. But this is all you can demonstrate to me.
You can never demonstrate that these circular lines pass at this point between the first circle and the tangent.
A secant A G may be shorter than another secant A GH:-granted; but it does not thence follow that your curve lines can pass between two lines which touch. They can pass, the master will reply, because the secant G H as distinguished from the secants A G, and AGH may be an “infiniment-petit” of the second order.
I do not understand what "an infiniment-petit" is, says the child; and the master is obliged to acknowledge that he understands it no more than his pupil. Here Malezieux, in his Elements of Geometry, bursts into an extacy He says positively, that there are incompatible truths. Would it not have been more simple to have said, that these lines have but one common point, on each side of which they separate.
I can always divide a number in thought; but does it thence follow that the number is infinite? Newton, in his integral, and in his differential calculation, does not use this great word; and Clairaut takes good care not to teach in his Elements of Geometry, that a hoop, may be passed between a ball and the table on which it lies. A careful distinction should be made between useful and curious geometry.
To the useful we owe the proportional compasses, invented by Galileo, the measurement of triangles, that of solids, and the circulation of moving forces. Most other problems may enlighten and strengthen the intellect; very few of them will be of sensible utility to mankind. Square curves as long as you like--and while displaying extreme sagacity only resemble an arithmetician who examines the properties of his numbers, instead of calculating the amount of his own property.
When Archimedes found the specific weight of bodies, he rendered a service to mankind : what service will you render by finding three numbers, so as that the difference of the squares of two of them, added to the cube of the three, will still be a square, and that the sum of the three differences added to the same cube, shall make another square ? Nugæ difficiles."*
* In geometry, as in most sciences, it is very rare that an isolated proposition is of immediate utility. But the theories most useful in practice are formed of propositions which curiosity GLORY-GLORIOUS.
GLORY is reputation joined with esteem, and is complete when admiration is superadded. It always supposes that which is brilliant in action, in virtue, or in talent, and the surmounting of great difficulties. Cæsar, Alexander, had glory. The same can hardly be said of Socrates. He claims esteem, reverence, pity, indignation against his enemies; but the term glory applied to him would be improper ; his memory is venerable rather than glorious. Atilla had much brilliancy, but he has no glory; for history, which
may be mistaken, attributes to him no virtues : Charles XII. still has glory; for his valour, his disinterestedness, his liberality, were extreme. Success is sufficient for reputation, but not for glory. The glory of Henry IV. is every day increasing; for time has brought to light all his virtues, which were incomparably greater than his defects.
Glory is also the portion of inventors in the fine arts; imitators have only applause. It is granted too to great talents, but in sublime arts only. We may well say, the glory of Virgil, or of Cicero, but not of Martial, nor of Aulus Gellius.
Men have dared to say, the glory of God: God created the world for his glory; not that the Supreme Being can have glory; but that inen, having no expressions suitable to him, use for him those by which they are themselves most flattered.
Vain glory is that petty ambition which is contented with appearances, which is exhibited in pompous dis
alone brought to light, and which long remained useless without its being possible to divine in what way they should one day cease to be so. In this sense it may be said, that in real sciences, no theory, no research, is in effect useless.-French Ed.
It was by taking up some minute neglected remainders that La Place, in his Mecanique Céleste, has cleared up several apparent doubts and anomalies in the Newtonian system of planetary revolutions.-T.
play, and never elevates itself to greater things. Sovereigns, having real glory, have been known to be nevertheless fond of vain glory--seeking too eagerly after praise, and being too much attached to the trappings of ostentation.
False glory often verges towards vanity; but it often leads to excesses, while vain glory is more confined to splendid littlenesses. A prince who should look for honour in
revenge, would seek a false glory rather than a vain one.
To give glory, signifies to acknowledge, to bear witness. Give glory to truth, means acknowledging truth—Give glory to the God whom you serve-Bear witness to the God whom you serve,
Glory is taken for heaven-He dwells in glory; but this is the case in no religion but ours. It is not allowable to say that Bacchus, or Hercules, was received into glory, when speaking of their apotheosis.
The saints and angels have sometimes been called the glorious, as dwelling in the abode of glory.
Gloriously is always taken in the good sense; he reigned gloriously; he extricated himself gloriously from great danger or embarrassment.
To glory in, is sometimes taken in the good, sometimes in the bad sense, according to the nature of the object in question. He glories in a disgrace which is the fruit of his talents and the effect of
We say of the martyrs, that they glorified God--that is, that their constancy made the God whom they attested revered by men.
That Cicero should love glory, after having stifled Catiline's conspiracy, may be pardoned him.
That the king of Prussia, Frederic the Great, should have the same feelings after Rosbach and Lissa, and after being the legislator, the historian, the poet, and the philosopher of his country—that he should be passionately fond of glory, and at the same time, have self-command enough to be modestly so- he will, on that account, be the more glorified.
That the empress Catherine II. should have been forced by the brutal insolence of a Turkish sultan to display all her genius; that from the far north she should have sent four squadrons which spread terror in the Dardanelles and in Asia Minor; and that, in 1770, she took four provinces from those Turks who made Europe tremble ;-she will not be reproached with enjoying her glory, but will be admired for speaking of her successes with that air of indifference and superiority, which shows that they were merited.
In short, glory befits geniuses of this sort, though belonging to the very mean race of mortals.
But if, at the extremity of the west, a townsman of a place called Paris thinks he has glory in being harrangued by a teacher of the university, who says to him, “Monseigneur, the glory you have acquired in the exercise of your office, your illustrious labours with which the universe resounds, &c. then I ask if there are mouths enow in that universe to celebrate, with their hisses, the glory of our citizen, and the eloquence of the pedant who attends to bray out this harangue at monseigneur's hotel?
We are such fools, that we have made God glorious like ourselves.
That worthy chief of the dervises, Ben-al-Betif, said to his brethren one day :
“ My brethren, it is good that you should frequently use that sacred formula of our koran— In the name of the most merciful God;' because God uses mercy, and
you learn to do so too, by often repeating the words that recommend virtue, without which there would be few men left upon the earth. · But, my brethren, beware of imitating those rash ones who boast; on every occasion, of labouring for the glory of God.
“ If a young simpleton maintains a thesis on the categories, an ignoramus in furs presiding, he is sure to write in large characters, at the head of his thesis, ' Ek alha abron doxa.'- Ad majorem Dei gloriam.'-To the greater glory of God. If a good mussulman has had his house whitewashed, he cuts this foolish inscription in the door. A saka carries water for the greater glory