smallest rivulet, the hills, the woods, in their true places. Before that time, so confused was the topography, that on the eve of the battle of Fontenoi, the maps of the country being all examined, every one of them was found entirely defective. If a positive order had been sent from Versailles to an inexperienced general to give battle, and post himself as appeared most advisable from the maps, as sometimes happened in the time of the minister Chamillars, the battle would infallibly have been lost. A general who should carry on a war in the country of the Morlachians, or the Montenegrians, with no knowledge of places but from the maps, would be at as great a loss as if he were in the heart of Africa. Happily, that which has often been traced by geographers, according to their own fancy, in their closets, is rectified on the spot. In geography, as in morals, it is very difficult to know the world without going from home. It is not with this department of knowledge as with the arts of poetry, music, and painting. The last works of these kinds are often the worst. But in the sciences, which require exactness rather than genius, the last are always the best, provided they are done with some degree of care. One of the greatest advantages of geography, in my opinion, is this :-Your fool of a neighbour, and his wife almost as stupid, are incessantly reproaching you with not thinking as they think in the rue St. Jacques. See,” say they, “ what a multitude of great men have been of our opinion, from Peter the Lombard down to the abbé Petit-pied. The whole universe has received our truths; they reign in the faubourg St. Honoré, at Chaillot and at Etampes, at Rome and among the Uscoques.” Take a map of the world; show them all Africa, the empires of Japan, China, India, Turkey, Persia, and that of Russia, more extensive than was the Roman empire; make them pass their finger over all Scandinavia, all the north of Germany, the three kingdoms of Great Britain, the greater part of the Low Countries, and of Helvetia ; in short make them observe, in the four great divisions of the earth; and in the fifth, which is as little known as it is great in extent, the prodigious number of races, who either never heard of those opinions, or have combated them, or have held them in abhorrence, and you' will thus oppose the whole universe to the rue St. Jacques. You will tell them that Julius Cæsar, who extended his power much further than that street, did not know a word of all which they think so universal; and that our ancestors, on whom Julius Cæsar bestowed the lash, knew no more of them than he did. They will then, perhaps, feel somewhat ashamed at having believed that the organ of St. Severin's church gave the tone to the rest of the world. GEOMETRY. The late M. Clairaut conceived the idea of making young people learn the elements of geometry with facility. He wished to go back to the source, and to trace the progress of our discoveries and the occasions which produced them. This method appears agreeable and useful; but it has not been followed. It requires in the master a flexibility of mind which knows how to adapt itself, and an accommodating spirit which is rare among those who follow the routine of their profession. It must be acknowledged that Euclid is somewhat unattractive; a beginner cannot divine whither he is to be led. Euclid says, in his first book, that “ if a straight line is divided into two equal and into two unequal parts, the squares of the unequal segments are double of the squares of half the line, and of the portion of it included between the points of intersection.” A diagram is necessary to understand this obscure theorem; and when it is understood, the student says, -Of what service can it be to me? what does it matter?-He is disgusted with a science, of which he does not soon enough perceive the utility. Painting began with the desire of roughly sketching on a wall the features of some one dear to the designer. Music, before the octave was found, was a rude mixture of some sounds which were pleasing to the ear. The setting of the stars was observed before men became astronomers. And it appears that the course of beginners in geometry should be similarly guided. I will suppose that a child of ready conceptions hears his father say to his gardener, “ you will plant tulips ơn this flower-bed half a foot from one another.” The child wishes to know how many tulips there will be. He runs to the flower-bed with his tutor. The parterre is inundated, and only one side of the flower-bed appears. This side is thirty feet long; but the breadth is not known. The master in the first place easily makes him understand that these tulips must border the parterre at the distance of six inches from one another. Here are already sixty tulips for the first row on that side. There are to be six lines. The child sees that there will be six times sixty, or three hundred and sixty tulips. But what will be the breadth of this bed, which I cannot measure? It will evidently be six times six inches, which are three feet. He knows the length and the breadth. He also wishes to know the superficies. Is it not true, his teacher asks him, that if you were to run a rule three feet long and one foot broad over this bed, from one end to the other, it would successively have covered the whole? Here, then, we have the superficies; it is three times thirty. This piece of ground is ninety square feet. A few days after, the gardener stretches a cord lengthwise from one angle to the other; which cord divides the rectangle into two equal parts. This, says the pupil, is the same length as one of the two sides. TUTOR. No. It is longer. PUPIL. How? If I pass a line over this cross-line, which you call a diagonal, it will be no longer than the two others.-- When I form the letter N, is not this line, which joins the two straight strokes together, of the same height as they are? TUTOR. It is of the same height, but not of the same length; that is demonstrated.-Bring down this diagonal to one of the sides, and you will find that it exceeds it. PUPIL. And by how much precisely does it exceed it? TUTOR. There are cases in which this can never be known; as it will never be known precisely what is the square root of five. PUPIL. But the square root of five is two and a fraction. TUTOR. But this fraction cannot be expressed in figures, since the square of a number composed of a whole number and a fraction cannot be a whole number. So, in geometry, there are lines, the relations of which cannot be expressed. PUPIL. Here, then, is a difficulty in my way.—What! shall I never know my accompts? Is there, then, nothing certain ? TUTOR. It is certain that this sloping line divides the quadrangle into two equal parts ; but it is no more surprising that this small remainder of the diagonal line has not a common measure with the sides, than that in arithmetic you cannot find the square root of five. You will not therefore the less know your accompts: for if an arithmetician tells you that he owes you the square root of five crowns, you have only to reduce these five crowns into smaller pieces; as, for instance, into liards, and will have 'twelve hundred of them; the square root of which is between thirty-four and thirty-five : so that you will make your reckoning within a liard. Nothing must be made a mystery in arithmetic or in geometry. These first openings sharpen the young man's wit. His 'master having told him that the diagonal of a square is incommensurable—not measurable by the sides and the base, informs him that with this line, the value of which can never be known, he will nevertheless produce a square which shall be demonstrated to be double of any given square. For this purpose, he first shows him that the two triangles which divide the square are equal, and then, by tracing a very simple figure, leads him to a comprehension of the famous theorem which Pythagoras found established among the Indians, and which was known to the Chinese—that any figure constructed on the larger side of a right-angled triangle is equal to the two similar figures constructed on the other sides. If the young man wishes to measure the height of a tower, or the breadth of a river which he cannot approach, each theorem immediately has its application; and he learns geometry practically. If he had merely been told that the product of the extremes is equal to the product of the means, he would have found this nothing more than a sterile problem: but he knows that the shadow of this stick is to the height of the stick as the shadow of the neighbouring tower is to the height of the tower. If, then, the stick be five feet, and its shadow one, and the shadow of the tower is twelve feet, he says, as one is to five, so is twelve to the height of the tower; then it is sixty feet. He wants to know the properties of a circle. He knows that the exact measure of its circumference cannot be had. But this extreme exactness is unnecessary in practice. The unrolling of a circle is its measurement. He will know that, this circle being a sort of polygon, its area is equal to a triangle, the short side of which is the radius of the circle, and its base the measure of the circumference. The circumferences of circles are to one another as their radii. Circles having the general properties of all similar rectilinear figures, and these figures being to one ano |