magnitudes together. Some quantities may be variable by themselves alone, while those connected with them are constant: as the abscisses of a parallelogram, whose ordinates may be considered as all equal, and therefore constant. The diameter of a circle and the parameter of a conic section are constant, while their ab scisses are variable. Variable quantities (see FLUXIONS) are usually denoted by the last letters of the alphabet 2, y, x, while the constant ones are denoted by the first letters a, t, c.

VARIANCE, in law, signifies any alteration of a thing formerly laid in a plea, or where the declaration in a cause differs from the writ, or from the deed upon which it is grounded. If there be a variance between the declaration and the writ, it is error, and the writ shall abate; and if there appear to be a material variance between the matter pleaded and the manner of pleading it, this is not a good plea, for the manper and matter of pleading ought to agree in substance, or there will be no certainty in it. Cro. Jac. 479.

VARIATION. s. (variatio, Latin.) 1. Change; mutation; difference from itself (Bentley). 2. Difference; change from one to another (Woodward). 3. Successive change (Shakspeare). 4. (In grammar.) Change of termination of nouns (Watts). 5. Change in natural phenomenons (Wotton). 6. Deviation (Dryden).

VARIATION, in astronomy. The variation of the moon, called by Bulliald the reflection of her light, is the third inequality observed in the moon's motion; by which, when out of the quadratures, her true place differs from her place twice equated. See ASTRONOMY.

Newton makes the moon's variation to arise partly from the form of her orbit, which is an ellipsis; and partly from the inequality of the spaces which the moon describes in equal times, by a radius drawn to the earth.

To find the greatest variation.-Observe the moon's longitude in the octants; and to the time of observation compute the moon's place twice equated; then the difference between the computed and observed place is the greatest variation (35′ 41′′·6) sin. 2 ( ) −O).

Tycho makes the greatest variation 40′ 30′′; and Kepler makes it 51′ 49′′. But Newton makes the greatest variation, at a mean distance between the sun and the earth, to be 35′ 10′′; at the other distances, the greatest variation is in a ratio compounded of the duplicate ratio of the times of the moon's 'synodical revolution directly, and the triplicate ratio of the distance of the sun from the earth inversely. And therefore in the sun's apogee, the greatest variation is 33′ 14′′, and in his perigee 37' 11"; provided that the eccentricity of the sun is to VOL. XI.-PART II.

Sir Isaac Newton makes the index of the inequality, or variation of curvature, to be the ratio of the fluxion of the radius of curvature to the fluxion of the curve itself: and Maclaurin, to avoid the perplexity that different notions, connected with the same terms, occasions to learners, has adopted the same definition; but he suggests, that this ratio gives rather the va riation of the ray of curvature, and that it might have been proper to have measured the variation of curvature rather by the ratio of the fluxion of the curvature itself to the fluxion of the curve; so that, the curvature being inversely as the radius of curvature, and consequently its fluxion as the fluxion of the radius itself di rectly, and the square of the radius inversely, its variation would have been directly as the measure of it according to Newton's definition, and inversely as the square of the radius of curvature.

According to this notion, it would have been measured by the angle of contact contained by the curve and circle of curvature, in the same manner as the curvature itself is measured by the angle of contact contained by the curve and tangent. The reason of this remark may appear from this example: the variation of curvature, according to Newton's explication, is uniform in the logarithmic spiral, the fluxion of the radius of curvature in this figure being always in the same ratio to the fluxion of the curve; and yet, while the spiral is produced, though its curvature decreases, it never vanishes; which must appear a strange paradox to those who do not attend to the import of sir Isaac Newton's definition.

The variation of curvature at any point of a conic section is always as the tangent of the angle contained by the diameter that passes through the point of contact, and the perpendicular to the curve at the same point; or to the angle formed by the diameter of the section, and of the circle of curvature. Hence the variation of curvature vanishes at the extremities of either axis, and is greatest when the acute angle, contained by the diameter passing through the point of contact and the tangent, is least.

When the conic section is a parabola, the variation is as the tangent of the angle, contained by the right line drawn from the point of contact to the focus, and the perpendicular to the curve. See CURVATURE.

From sir Isaac Newton's definition may be

D

VARIGNON (Peter), a celebrated French mathematician and priest, was born at Caen, in 1654, and died suddenly in 1722, at 68 years of age. He was the son of an architect in middling circumstances, but had a college education, being intended for the church. An accident threw a copy of Euclid's Elements in his way, which gave him a strong turn to that kind of learning. The study of geometry led him to the works of Des Cartes on the same science, and there he was struck with that new light which has from thence spread over the world.

He abridged himself of the necessaries of life to purchase books of this kind, or rather considered them of that number, as indeed they ought to be. What contributed to heighten this passion in him was, that he studied in private: : for his relations observing that the

books he studied were not such as were commonly used by others, strongly opposed his application to them. As there was a necessity for his being an ecclesiastic, he continued his theological studies, yet not entirely sacrificing his favourite subject to them.

At this time the abbé St. Pierre, who studied philosophy in the same college, became acquainted with him. A taste in common for rational subjects, whether physics or metaphysics, and continual disputations, formed the bonds of their friendship. They were mutually serviceable to each other in their studies. The abbé, to enjoy Varignon's com pany with greater ease, lodged him with him self; thus, growing still more sensible of his merit, he resolved to give him a fortune, that be might fully pursue his genius, and improve his talents; and, out of only 1800 livres a year, which he had himself, he conferred 300 of them upon Varignon.

The abbé, persuaded that he could not do better than go to Paris to study philosophy, settled there in 1686, with M. Varignon, in the suburbs of St. Jacques. There each studied in his own way; the abbé applying himself to the study of men, manners, and the principles of government; whilst Varignon was wholly occupied with the mathematics.

I, says Fontenelle, who was their country. man, often went to see them, sometimes spending two or three days with them. They had also room for a couple of visitors, who came from the same province. We joined together with the greatest pleasure. We were young, full of the first ardour for knowledge, strongly united, and, what we were not then perhaps disposed to think so great a happiness, lute known. Varignon, who had a strong constitution, at least in his youth, spent whole days in study, without any amusement or recreation, except walking sometimes in fine weather. I have heard him say, that in studying after supper, as he usually did, he was often surprised to hear the clock strike two in the morning; and was much pleased that four hours rest were sufficient to refresh him. He did not leave his studies with that heaviness which they usually create; nor with that

weariness that a long application might occa sion. He left off gay and lively, filled with pleasure, and impatient to renew it. In speaking of mathematics, he would laugh so freely, that it seemed as if he had studied for diversion. No condition was so much to be envied as his; his life was a continual enjoyment, delighting in quietness.

In the solitary suburb of St. Jacques he formed, however, a connexion with many other learned men; as Du Hamel, Du Verney, De la Hire, &c. Du Verney often asked his assistance in those parts of anatomy connected with mechanics they examined together the positions of the muscles and their directions; hence Varignon learned a good deal of anatomy from Du Verney, which he repaid by the application of mathematical reasoning to that subject.

At length, in 1687, Varignon made himself known to the public by a Treatise on New Mechanics, dedicated to the Academy of Sciences. His thoughts on this subject were, in effect, quite new. He discovered truths, and laid open their sources. In this work he demonstrated the necessity of an equilibrium, in such cases as it happens in, though the cause of it is not exactly known. This discovery Varignon made by the theory of compound motions, and is what this essay turns upon.

This new Treatise on Mechanics was greatly admired by the mathematicians, and procured the author two considerable places, the one of geometrician in the Academy of Sciences, the other of professor of mathematics in the college of Mazarine, to which he was the first person raised.

Varignon catched eagerly at the science of infinitesimals as soon as it appeared in the world, and became one of its niost early cultivators. When that sublime and beautiful method was attacked in the academy itself (for it could not escape the fate of all innovations), he became one of its most zealous defenders, and in its favour he put a violence upon his natural character, which abhorred all conten tion. He sometimes lineated that this dispute had interrupted him in his enquiries into the integral calculation so far, that it would be difficult for him to resume his disquisition where he had left it off. He sacrificed infinitesimals to the interest of infinitesimals, and gave up the pleasure and glory of making a farther progress in them when called upon by duty to undertake their defence.

All the printed volumes of the academy bear witness to his application and industry. His works are never detached pieces, but complete theories of the laws of motion, central forces, and the resistance of mediums to motion. In these he makes such use of his rules, that nothing escapes him that has any connexion with the subject he treats.

Geometrical certainty is by no means incompatible with obscurity and confusion, and those are sometimes so great that it is surprising a mathematician should not miss his way in so

Histoire des Revolutiones arrivés en Europe en matiere de Religion, 1616: he died in 1696: he is a very unfair historian, especially in matters where religion is concerned.

VARIOLA. (from varius, changing colour, because it disfigures the skin.) The small pox. A genus of disease in the class pyrexia and order exanthemata of Cullen; distinguished by synocha; eruption of red pimples on the third day, which on the eighth day contain puss, and drying, fall off in crusts.

It is a disease of a very contagious nature, supposed to have been introduced into Europe from Arabia, and in which there arises a fever, that is succeeded by a number of little inflammations in the skin, which proceed to suppuration, the matter formed thereby being capable of producing the disorder in another person. It makes its attack on people of all ages, but the young of both sexes are more liable to it than those who are much advanced in life; and it may prevail at all the seasons of the year, but in general is most prevalent in the spring and summer.

The small-pox is distinguished into the distinct and confluent, implying that in the former the eruptions are perfectly separate from each other, and that in the latter they run much into one another.

Both species are produced either by breathing air impregnated with the effluvia arising from the body of those who labour under the disease, or by the introduction of a small quantity of the variolous matter into the habit by inoculation; and it is probable that the difference of the small-pox is not owing to any difference in the contagion, but depends on the state of the person to whom it is applied, or on certain circumstances concurring with the application of

it.

A variety of opinions have been entertained respecting the effect of the variolous infection on the fetus in utero; a sufficient number of instances, however, has been recorded, to ascertain that the disease may be communicated from the mother to the child. In some cases, the body of the child at its birth has been covered with pustules, and the nature of the disease has been most satisfactorily ascertained by inoculating with matter taken from the pustules. In other cases, there has been no appearance of the disease at the time of the birth, but an eruption and other symptoms of the disease have appeared so early, as to ascertain that the infection must have been received previously to the removal of the child from the uterus. Four different states or stages are to be observed in the small-pox: first, the febrile; second, the eruptive; third, the maturative; and fourth, that of declination or scabbing, which is usually known by the name of secondary fever. When the disease has arisen naturally, and is of the distinct kind, the eruption is commonly preceded by a redness in the eyes, soreness in the throat, pains in the head, back, and loins, weariness and faintness, alternate fits of chillness and heat, thirst, nausea, inclination to vomit, and a quick pulse.

In some instances, these symptoms prevail in a high degree, and in others they are very moderate and trifling. In very young chil dren, startings and convulsion are apt to take place a short time previous to the appearance of the eruption, always giving great alarm to those not conversant with the frequency of the occurrence.

About the third or fourth day from the first seizure, the eruption shews itself in little red spots (similar to flea-bites) on the face, neck, and breast, and these continue to increase in number and size for three or four longer, at the end of which time, they are to be observed dispersed over several parts of the body.

If the pustules are not very numerous, the febrile symptoms will generally go off on the appearance of the eruption, or they will become very moderate. It sometimes happens, that a number of little spots of an erysipelatous nature are interspersed amongst the pustules; but these generally go in again as soon as the suppuration commences, which is usually about the fifth or sixth day, at which period, a small vesicle containing an almost colourless fluid may be observed upon the top of each pimple. Should the pustules be perfectly distinct and separate from each other, the suppuration will probably be completed about the eighth or ninth day, and they will then be filled with a thick yellow matter; but should they run much into each other, it will not be completed till some days later.

When the pustules are very thick and numerous on the face, it is apt about this time to become much swelled, and the eyelids to be closed up, previous to which, there usually arises a hoarseness, and difficulty of swallowing, accompanied with a considerable discharge of viscid saliva. About the eleventh day, the swelling of the face usually subsides, together with the affection of the fauces, and is succeeded by the same in the hands and feet, after which the pustules break, and discharge their contents, and then becoming dry, they fall in crusts, leaving the skin which they covered of a brown red colour, which appearance continues for many days. In those cases where the pustules are large, and are late in becoming dry and falling off, they are very apt to leave pits behind them; but where they are small, suppurate quickly, and are few in num ber, they neither leave any marks behind them, nor do they occasion much affection of the system.

In the confluent small-pox, the fever which precedes the cruption is much more violent than in the distinct, being attended usually with great anxiety, heat, thirst, nausea, vomit. ing, and a frequent and contracted pulse, and often with coma or delirium. In infants, convulsive fits are apt to occur, which either prove fatal before any eruption appears, or they usher in a malignant species of the disease.

The eruption usually makes its appearance about the third day, being frequently preceded or attended with a rosy efflorescence, similar to what takes place in the measles; but the