Parallelogram

“ All of you who have studied mechanics know very well what a "resultant" means. You know that when a body is acted on by two forces at the same time, one force carrying it in this direction, and another force in that direction, we want to know in what direction it will go, and how far it will go. To arrive at this we simply complete what is called the parallelogram of forces. In fact, it is just as if a body were acted on at two different times, by a force driving it in one direction, and then by a force driving it in the other direction. ”

“ Who can tell whether the parallelogram, which in our ignorance we have defined and drawn, and the whole of whose properties we profess to know, may not be all the while panting for exterior angles, sympathetic with the interior, or sullenly repining at the fact that it cannot be inscribed in a circle? ”

“ Parallelograms with are equiangular to one another have to one another the ratio which is compounded of the ratios of their sides. Let the parallelogram AC be equiangular to the parallelogram CF, having the angle BCD equal to the angle ECG: the parallelogram AC shall have to the parallelogram CF the ratio which is compounded of the ratios of their sides. ”

“ Looked at in this way, the true relationship between the two parallelogram-theorems is seen to be the very opposite of the one held with conviction by scientific thinking up to now. Instead of the parallelogram of forces following from the parallelogram of movements, and the entire science of dynamics from that of kinematics, our very faculty of thinking in kinematic concepts is the evolutionary product of our previously acquired intuitive experience of the dynamic order of the world. ”

“ A square is more beautiful than a parallelogram. The former exceeds the latter in regularity and in uniformity of parts. But this holds with respect to intrinsic beauty only; for in many instances, utility comes in to cast the balance on the side of the parallelogram. This figure for the doors and windows of a dwelling-house, is preferred because of utility; and here we find the beauty of utility prevailing over that of regularity and uniformity.
A parallelogram again depends, for its beauty, on the proportion of its sides. The beauty is lost by a great inequality of sides. ”

“ In the primary parallelogram Π0, however, ƒ (z) is only infinite at z = 0 in the neighbourhood of which its expansion is of the form z−2 + (power series in z) . Thus ƒ (z) is such a doubly periodic function as was to be constructed, having in any parallelogram of periods only one pole, of the second order.
It can be shown that any single valued meromorphic function of z with ω and ω′ as periods can be expressed rationally in terms of ƒ (z) and φ (z) , and that [φ (z) ] 2 is of the form 4 [ƒ (z) ] 3 + Aƒ (z) + B,
where A, B are constants. ”

“ A parallelogram and a rhomb are less regular than a square: the parallelogram is subjected to no rule as to the length of sides, other than that the opposite sides be equal: the rhomb is subjected to no rule as to its angles, other than that the opposite angles be{390} equal. For the same reason, the circumference of an ellipse, the form of which is susceptible of much variety, is less regular than that of a circle.
23. Regularity, properly speaking, belongs, like beauty, to objects of sight: like beauty, it is also apply’d figuratively to other objects. ”

“ While I was glad to use examples and models to instruct my pupils, I wished them to grasp certain geometrical relations through exercise of imagination. They had long known that the area of a parallelogram is the product of its base and height; they were now required to conceive that any triangle has half the area of a parallelogram of equal height and base. ”

“ Pantographs have been made in many shapes unnecessary to describe, as they are all of one principle—that of a parallelogram jointed at the four corners; the principal difference being in the position of the points and fulcrum in relation to the parallelogram. One thing is essential in every construction,—that is, that the fulcrum, tracer, and pencil should always be in a true line when the instrument is set for use. ”

“ In any parallelogram (AC) , every two parallelograms (AF, FC) which are about a diagonal are similar to the whole and to one another.
Dem.—Since the parallelograms AC, AF have a common angle, they are equiangular [I. xxxiv.] , and all that is required to be proved is, that the sides about the equal angles are proportional. Now, since the lines EF, BC are parallel, the triangles AEF, ABC are equiangular; therefore [iv.] AE : EF :: AB : BC, and the other sides of the parallelograms are equal to AE, EF; AB, BC: hence the sides about the equal angles are proportional ”

“ The best way to introduce this theorem is to cut a parallelogram from paper, and then, with the class, separate it into two parts by a cut perpendicular to the base. The two parts may then be fitted together to make a rectangle. In particular, if we cut off a triangle from one end and fit it on the other, we have the basis for the proof of the textbooks. The use of squared paper for such a proposition is not wise, since it makes the measurement appear to be merely an approximation. The cutting of the paper is in every way more satisfactory. ”

“ A regular pentagon may be cut into as few as six pieces that will fit together without any turning over and form a square, as I shall show below. Hitherto the best answer has been in seven pieces—the solution produced some years ago by a foreign mathematician, Paul Busschop. We first form a parallelogram, and from that the square. The process will be seen in the diagram on the next page. ”

“ If, in solving a dynamical problem, a parallelogram is drawn, of which the sides and diagonal represent forces, and by putting magnitudes of extension for magnitudes of force a measurable relation is established between quantities not else to be dealt with; it may be fairly said that geometry plays towards mechanics much the same part that the fire of the founder plays towards the metal he is going to cast. ”

“ The relative motion of the point P will be the same as in the real case.
During this motion the element QQ' will generate an area in the form of a parallelogram whose sides are parallel and equal to QQ' and PP'. ”

“ Such expressions as infinite sphere, radius, parallelogram, line, and so forth, are self-contradictory. A sphere is limited by its 181 own periphery, and a radius by the centre and circumference of its circle. A parallelogram of infinite altitude is impossible, because the limit of its altitude is assigned in the side which must be parallel to its base in order to constitute it a parallelogram. ”

“ Of all parallelograms which can be formed with diameters of given lengths the rhombus is the greatest.
452. Shew from I. 18 and I. 32 that if the hypotenuse BC of a right-angled triangle ABC be bisected at D, then AD, BD, CD are all equal.
453. If two equal straight lines intersect each other any where at right angles, the quadrilateral formed by joining their extremities is equal to half the square on either straight line.
454. Inscribe a parallelogram in a given triangle, in such a manner that its diagonals shall intersect at a given point within the triangle. ”

“ Cases might be multiplied indefinitely by placing various amounts of weight on the hooks, constructing the parallelogram on cardboard, and comparing it with the cords as before. We shall, however, confine ourselves to one more illustration, which is capable of very remarkable applications. ”

“ If we draw at a point P (fig. 28) on an hyperbola a tangent, the part HK between the asymptotes is bisected at P. The parallelogram PQOQ′ formed by the asymptotes and lines parallel to them through P will be half the triangle OHK, and will therefore be constant. If we now take the asymptotes OX and OY as oblique axes of co-ordinates, the lines OQ and QP will be the co-ordinates of P, and will satisfy the equation xy = const. = a2. ”

“ We understand the simple parallelogram by which the physicist represents to us the action of two forces at incidence, we may represent to ourselves the motion of any one of the heavenly bodies as the resultant of the centrifugal and centripetal forces, but when we come to consider the formation of a crystal, and watch the regularity of shape and grouping, this very uniformity which had been before an explanation now seems all at once to represent an insoluble mystery separating the process forever from those others. ”

“ But a reasoner saves himself all this trouble by seeing that it is the essence (pro hac vice) of a triangle to be the half of a parallelogram whose area is the height into the entire base. To see this he must invent additional lines; and the geometer must often draw such to get at the essential property he may require in a figure. The essence consists in some relation of the figure to the new lines, a relation not obvious at all until they are put in. The geometer's sagacity lies in the invention of the new lines. ”

“ Similarly a mathematician should know that a rhombus is so called from its resemblance to a form of spinning-top (ῥομβος from ῥεμβω, to spin) and that, just as a parallelogram is a figure formed by two pairs of parallel straight lines, so a parallelepiped is a solid figure bounded by three pairs of parallel planes (παραλληλος, parallel, and επιπεδος, plane) ; incidentally, in the latter case, he will be saved from writing ‘parallelopiped’, a monstrosity which has disfigured not a few textbooks of geometry. Another good example is the word hypotenuse ”