they are to the sentiment of L., may deserve examination. But one thing is particularly deserving of notice in these notations, viz. a as ev, which is to be demonstrated: whence it seems that L. not even then, after 16 years, had not found out a demonstration of the supposition formerly put off to Bernoulli; viz, that an action performing any thing in a single time, is double of an action performing the same thing in a double time: since an action performing any thing in a single time, does it with twice the velocity of an action performing the same thing in a double time. But how W. demonstrates this, we shall examine presently.

DEFINITIONS. Def. 1.-I call that, with Leibnitz, vis viva, or merely vi or force, which adheres to local motion. 2. A pure force, is that which is not resisted in acting by any contrary force. Hence such a pure force remains unvaried in the whole time of action, and is not in the least exhausted by the effect it produces. 3. A pure action, is that which is exercised by a pure moving force. Such as the action of a moveable carried with an equable motion in an unresisting medium. 4. A uniform action, is that which is double in a double time, triple in a triple, &c. or in general which is as the time; or such an action as obtains in equable motion. 5. The effect of a moving force beyond the conflict, is the translation of a moveable through a space.

AXIOMS. Axiom 1.-If two or more equal moveables be moved with equal. celerity, their force is the same. 2. The same action is performed by the same force in the same time. That a greater action is performed by the same force in a longer time than in a shorter, and that a greater action is performed in the same time by a greater force than by a less, no one doubts. Therefore the quantity of an action depends on the quantity of force and time. So that if the forces be equal, and the time the same, the action also must be the same. moveable be transferred through the same space, the effect is the same.

3. If the same

THEOREMS. Theor. 1.-When unequal bodies are moved with the same velocity, the forces are as the masses. 2. Uniform actions, performed in the same time, are to each other as their forces. 3. Uniform actions, performed with equal forces, are to each other as the times in which they are performed. 4. Uniform actions are in the ratio compounded of the times and forces. 5. Unequal forces perform the same action in times reciprocally proportional to each other. 6. When two equal moveables are carried through unequal spaces, the effects are as the spaces. 7. When two moveables are carried through the same space, the effects are as the masses. 8. When two moveables are carried through any spaces, the effects are in the ratio compounded of the masses and spaces. 9. In equable motions, the effects are in a ratio compounded of the masses, velocities, and times. The demonstrations of these 9 theorems, about

which there is no controversy with the Leibnitzians, and which being the same with those given by Wolf, are here omitted.

Theor. 10.-Actions, by which the same effect is produced, are as the velocities. It is on this theorem that the whole matter turns: if it be true, the Leibnitzian doctrine is to be embraced; if not, it is to be rejected': therefore its demonstration must be scrupulously examined. It is divided by Wolf into 3 cases: but as the 2d and 3d depend on the first, we shall need only consider this one. Mr. W. says, "when moveables are equal, and the same effect is produced in different times, the velocities will be reciprocally as the times in which it is produced; that is, a body, which produces an effect in the time, is moved with the velocity 2v, when another, which produces the effect in the time t, is moved with the single velocity v; and so on. Now it is evident that a uniform action is double, which produces the same effect in half the time, triple which is subtriple, and so on."

But do you say, Mr. Wolf, that this is evident? What if I should deny it? What if I should say that any action, which produces the same effect, is the same, in whatever time it produces it? This is the very supposition of Leibnitz, of which, in his letter to Bernoulli in 1696, he says he has not discovered a method of demonstrating a priori, and in his letter to yourself in 1711, he says is still to be demonstrated. And yet you do not endeavour to demonstrate it, but say it is evident. Now I deny its being evident, and thus your demonstration falls to the ground, and the supposition along with it.

But before substituting a new one, let us consider a little what is understood by action, and what by effect. Wolf, after the example of Leibnitz, has omitted the definition of action. He only distinguishes what is a pure action, viz. that which is free from all impediment; and what is a uniform action, viz. that which increases in proportion to the time: but what he means by action itself is nowhere determined. But till this is done, nothing can be demonstrated, as Bernoulli advised Leibnitz long since.

If I might venture to supply this defect, I would ascribe the same definition to action, which Wolf has given of effect; since there seems to be no other difference between action and effect, than that action, if I may so speak, is an effect in fieri, and effect an absolute action, or one that is perfected. For in Wolf's example, a vis viva is that which transfers a moveable through a space; therefore the action of a vis viva is the translation of a moveable through a space; and the effect of a vis viva, is also the translation of a moveable through a space; or rather, an effect is a moveable already transferred through the same space.

But generally an action is the preceder of an effect or rather, an action is that by which any thing is effected; but an effect is the thing itself which is effected. For example, if I write a page, my action will be the writing of a page,

and the effect will be a page written. Or, if a workman whiten a wall, his action will be the whitening of a wall, and the effect will be a wall whitened, Or, if a labourer dig a garden, his action is the digging of a garden, and the effect is a garden digged.

Our Theor. 10.-Of equal actions, the effects are equal.

Let any vis viva a perform any action; and let there be supposed any other vis viva B. Now that the vis viva в may perform an action equal to that of the vis viva A, it is necessary that the vis viva в should act exactly as much as the vis viva a has acted. Therefore, after the completion of the action of в, as much will be acted by the force в, as has been acted by the force a; that is, vis viva в will be equal to the effect of the vis viva A, the actions of which were equal.

the effect of the

Our Theor. 11.-Actions are in proportion to the effects. For let the effect e be produced by the action a. Therefore another effect e, equal to the first, will, by theor. 10, be produced by another equal action a; consequently the effect twice e will be produced by the action twice a. In like manner it appears that the effect thrice e must be produced by the action thrice a, &c. And generally, that the effect ne (= E) must be produced by the action na (= A.) Therefore A: a :: E: e; that is, the actions are in the ratio of the effects. Our Theor. 12.-Forces are in the ratio compounded of the masses and velocities.

For, by theor. 4, actions are in the ratio compounded of the times and forces. And, by theor. 11, the actions are in the ratio of the effects. Therefore, the effects are in the ratio compounded of the times and forces. But, by theor. 8, effects are in the ratio compounded of the masses and spaces. Therefore the ratio compounded of the times and forces, is equal to the ratio compounded of the masses and spaces. Therefore the forces are in the ratio compounded of the masses and spaces directly, and of the times reciprocally; that is, in the ratio compounded of the masses and velocities.

Of Two Extraordinary Deers' Horns, found under-ground in different Parts of Yorkshire. By Mr. Tho. Knowlton N° 479, p. 124.

The head and horns here described, were found in a sand-bed, in the river Rye, which runs into the Derwent, in the east-riding, belonging to Ralph Crathorn, esq. They were discovered as he was fishing for salmon; the net happening to hang on 1 or 2 of the antlers, he ordered to pull away; by which some of the antlers were broken off, and discovered it to be part of a deer's horn. At length, with some difficulty, it was dug out pretty entire. Mr. Crathorn supposes, that these wild moors were once inhabited with this kind of deer, not any such now being known to be in this kingdom; and supposes it is, at least, 7 or

800 years since its death: and that by age or poverty destroyed, and by time buried in those sands. These horns had many antlers, and were about 34 feet in length, from the root to the tip.

Another skull, and horns, of the palmated kind, are here described also: the head and horns weighed together 4 st. 12 lb. or 68 lb.

The length of the skull, from the nose-end to the back-part

Yet it is evident the horns are not at their full growth, being yet covered with what is called the velvet.

They were found in a peat-moss, at Cowthorp near North Dreighton in Yorkshire, in the year 1744; and were dug up from the depth of 6 feet out of the peat moss,

But what he thinks more extraordinary is, that the late earl of Carlisle's steward, Mr. Joice, in digging the foundation of a house and cellars, found, at the depth of 6 feet, a part of a jaw-bone with teeth, and a horn of a buck, or stag of most exceedingly large dimensions, which lay buried under 2 feet of common soil; then one foot of scalping or sand-bed; then 18 inches of stone; then another vein of sand, 6 inches; then another head of stone; under which lay those before-mentioned jaw-bone, and piece of horn; which, in all appearance, to every one that viewed these stratums, had never been removed.

Dimensions of the Deers' Horns in the Museum of the Royal Society.

Length of the skull ....

Breadth of the forehead...

Length of each horn. ...

Distance of the extreme tips of the horns.. 6

The Phenomena of Venus, represented in an Orrery made by Mr. James Ferguson,* agreeable to the Observations of Seignior Bianchini. N° 479, p. 127. In all the common orreries, Venus is represented as having her axis perpendicular to the plane of the ecliptic, and her diurnal motion on it equal to 23 hours

Mr. Ferguson was a very uncommon genius, especially in mechanical contrivances and structures, for he executed many machines himself in a very neat manner. He had a good taste in astronomy, and generally in natural and experimental philosophy; he also possessed a happy manner of explanation in his lectures, in a clear, easy, and familiar way. Yet his general mathematical knowledge was little or nothing. Of algebra he understood little more than the notation; and he never could demonstrate a proposition in geometry; his constant method being to satisfy himself, as to the truth of any problem, with a very exact construction and measurement by scale and compasses.

of our terrestrial time. Hence, as her annual motion is performed in about 225 of our days, it will contain 234 of hers; consequently, to an eye placed in Venus, the sun will always appear to go through a sign of the zodiac in 19 of her days; and as her axis has no inclination, she must have a continual equality of her days and nights, without any variation of seasons, and so her annual motion can be of no other use than to keep her from falling down to the sun.

But Bianchini gives a very different account of her; which is, that her axis inclines 75° from the plane of her ecliptic; and that her diurnal motion is performed in 24 days and 8 hours of our time; and this will cause her year, which is equal to almost 225 of our days, to contain only 9+ of her days; and this odd quarter of a day in Venus will make every 4th year a leap-year to her, as happens to us on earth, by the 6 hours that our year contains above 365 days: and to her the sun will appear always to go through a sign of the zodiac in little more than of her day, which is equal to 184 of our days; and in going round the sun, her north pole constantly leans towards the 20th degree of Aquarius.

Thus, with regard to the absolute length of Venus's year, Bianchini agrees with Cassini and other astronomers: but differs widely in other very remarkable particulars, from which arise so many advantages, as to make that planet incomparably more fit for its inhabitants, than we could possibly conceive it to be by a quick rotation on an axis perpendicular to its annual path.

But, by such a motion as Bianchini describes, these inconveniences are avoided; for there is no place in Venus but what will have the 4 seasons every year, and the heated places will have time to cool; because, to any place over which the sun passes vertically on any given day, he will, on the next day, be 26 degrees from its vertex, even though the place be on the tropic; and if it be on the equator, one day's declination will remove him 37 degrees from it.

By passing a narrow slip of paper around the terrestrial globe, crossing the

Mr. F. was born in Bamfshire 1710, of poor parents. He first learned to read, by overhearing his father teach his elder brother; and indeed all his knowledge afterwards was of his own acquisition. He soon discovered a peculiar taste for mechanics, which first arose by seeing his father make use of a lever. He pursued this study to considerable length, while he was yet very young; and made a watch in wood-work, from having once seen one. At a fit age he went to service with a farmer; when tending the sheep gave him leisure to make considerable advances in mechanical and astronomical knowledge. He acquired also some skill in drawing; and going to Edinburgh, he there for some time followed the profession of taking portraits in miniature, at a small price: an employment by which he supported his family for several years, both in Scotland and England, while he pursued more serious studies, and the labour of philosophical lectures. Mr. F. was a F.R.S.; he died in 1776, at 66 years of age.

A full account of Mr. F.'s life was printed by himself, being prefixed to his Select Mechanical Exercises. An abstract of which may be seen in Dr. Hutton's Mathematical and Philosophical Dictionary, with a complete list of Mr F.'s separate publications; besides which, his numerous communications to the Royal Society are printed in the Philos. Trans. from vol. xliv. to vol. lvii. inclusive.