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The greatest line joining two points in the perimeter of a polygon is a side or a diagonal of the polygon.
For (fig. 2) PS is obviously less than one or other of the lines PQ, PR. Hence if AB, CD (fig. 3) are any two finite lines, L and M any points, one in each of these lines, LM is less than LC, or LD and LC is less than CA or CB. From this the theorem follows at once. The theorem and the above proof apply to crossed and gauche polygons.
Par un point fixe A d'une circonférence donnée on mène deux cordes AB et AC dont le produit a une valeur constante m2, puis on joint BC. Trouver 1° le lieu du pied D de la bissectrice de l'angle A du triangle ABC; 2° le lieu des centres des cercles inscrits et exinscrits à ce triangle.
Though every quantity, whatever be its nature, has magnitude, no quantity can be said to be large or small absolutely. When we speak of the size of any body we mean its size relatively to the size of some other body with which we compare it. A yard is large if we compare it with an inch; it is small when compared with a mile. In the former case the number which represents it is more than 60,000 times larger than the number by which it is represented in the latter case. A mere number is therefore useless as regards the statement of magnitude, except when accompanied by a clear indication of what the thing measured is compared with. The quantity in terms of which the comparison is made is called the unit, and the number which tells how often this unit is contained in a given quantity is called the numeric of that quantity.
It is well known that in a given triangle a one-fold infinity of triangles may be inscribed similar to a given triangle This becomes at once obvious on consideration of the converse problem; for we may circumscribe about a given triangle (A), a triangle similar to a second triangle (B), and having its sides parallel to the sides of (B).
Most of the following theorems occur in a more or less explicit form in text-books on the geometry of the parabola; but it will not, I hope, be without interest and value to consider them independently, and to prove them by using only the propositions of Euclid.
It is well known that the elliptic integral of the second kind may be represented by the arc of an ellipse, and mathematicians have sought with various success to represent similarly by the arc of an algebraic curve the elliptic integral of the first kind. The general solution of the problem has not been obtained, but Serret and Cayley have given solutions of a very general character.
Let represent any permutation of the first n natural numbers. If we take the first number, a1, and put it last, we get a new permutation; and if we perform this operation (n−1) times, we get the (n−1) permutations