determined by the same rule or equation, the ratio of the motions is determined in the same manner, when both are

supposed variable. These propositions are demonstrated strictly by the same method which is adopted for determining the fluxions of figures.

A TRIANGLE that has two of its sides given in position, is supposed to be generated by an ordinate moving parallel to itself along the base. When the base increases uniformly, the triangle increases with an accelerated motion, because its successive increments are trapezia, that continually increase. Therefore, if the motion with which the triangle flows was continued uniformly from any term for a given time, a less. space would be described by it than the increment of the triangle which is actually generated in that time by axiom 1., but a greater space than the increment which was actually generated in an equal time preceding that term, by axiom 2.; and hence it is demonstrated, that the fluxion of the triangle is accurately measured by the rectangle contained by the corresponding ordinate of the triangle, and the right linewhich measures the fluxion of the base. The increment which the triangle acquires in any time is resolved into two parts; that which is generated in consequence of the motion with which the triangle flows at the beginning of the time, and that which is generated in consequence of the acceleration of this motion for the same time. The latter is justly neglected in measuring that motion, or the fluxion of the triangle at that term, but may serve for measuring its acceleration, of the second fluxion of the triangle. The motion with which the triangle flows is similar to that of a body descending in free spaces by a uniform gravity, the velocity of which, at any term of the time, is not to be measured by the space described by the body in a given time, either before or after that term, because the motion continually increases, but by a mean between these spaces.

When the sides of a RECTANGLE increase or decrease with uniform motions, it may be always considered as the sum or difference of a triangle and trapezium; and its fluxion is derived from the last proposition. If the sides increase with uniform motions, the rectangle increases with an accelerated motion; and in measuring this motion at any term of the time, a part of the increment of the rectangle, that is here. determined, is rejected, as generated in consequence of the acceleration of that motion.

The fluxions of a CURVILINEAL AREA (whether it be generated by an ordinate moving parallel to itself, or by a ray

revolving about a given centre,) and of the solid, generated by the area revolving about the base, are determined by demonstrations of the same kind; and when the ordinates of the figure increase, the increment of the area is resolved in like manner into two parts, one of which is only to be retained in measuring the fluxion of the area, the other being rejected as generated in consequence of the acceleration of the motion with which the figure flows. An illustration of the second and third fluxions is given by resolving the increment of a pyramid or cone into the several respective parts that are conceived to be generated in consequence of the first, second, and third fluxions of the solid, when the axis is supposed to flow uniformly.

Some further Account of Polypi, in a Letter from the Duke of RICHMOND, F.R.S., to M. FOLKES, Esq. Pr. R. S. Dated Utrecht, May 24. (June 4.) 1743.

You will not be sorry to receive some further account of the polypus; and I must tell you what I have seen in M. Trembley's study at Sorgvliet. He has there 12 large large glasses, of about a foot high, each holding from a gallon to six quarts of water, all well stocked with those animals, to the amount of many hundreds. They are, in general, considerably larger than any I had before seen; and as I was first with him on a Tuesday, and made him a second visit on the Sunday following, I had the opportunity of seeing the prodigious increase they had made in those five days. Several single ones that I had left, had in that time put out five or six young ones each; and those I had seen him perform operations on, were not only recovered, but had most of them produced young ones also.

I saw him split the head of one about two o'clock in the afternoon, and at about seven the same evening, each head ate a small worm. I saw him split another from the head to the tail, and each of those parts also ate part of a worm before night.

Another operation I saw him make, which I had not before heard of, which was that by putting one of the points of a very small pair of sharp scissors into the mouth of a polypus, and forcing it out at the very end of the tail, he then laid it quite open like a pigeon, or a barbacute pig to be broiled; yet, in about five hours, I saw the same polypus with the parts so re-united again, that I could not perceive any thing had been done to it; and it then ate a worm larger than itself.

He then showed me another odd particular, which was one polypus that had fairly two heads, without any tail; that is, with a head at each end. This was an accidental production, and as follows: two young ones grew, as from one root, out of an old polypus. They both dropped off together, and their tails not being separated, they appeared as just mentioned; but, when I saw them, with several young ones putting out from their sides.

M. Trembley said he had seen the like sometimes before, but not often; and that they have then remained 10 or 12 days in that condition, after which they have separated. He had in one of his large glasses upwards of a hundred of these insects all full grown, and he regaled them all at once before me, with some thousands of small aquatic animalcules, not unlike fleas, of about the size of large ones, and which move about with great swiftness in the water. These were no sooner put in, but it was a curious and entertaining sight, to observe in how voracious a manner not only every polypus, but every young one also that had arms, though fixed to the side of its parent, seized and devoured these pucerons; and as the body of the polypus is transparent, every one made a very extraordinary appearance, from the number of pucerons in them; for in several I could very plainly, with my bare eye, distinguish and count five or six of them; and I could plainly discern some very small black spots, which I was assured were the eyes of these pucerons.

One extraordinary observation more of M. Trembley's is, that, in the double-headed polypus, there was at first but one common gut between them, so that the feeding of one head had the same effect as feeding them both. M. Trembley is particularly handy and dexterous in his operations, and explains himself about them with great exactness and perspicuity. He places some pieces of packthread across his glasses, towards the top to these some of the insects fix themselves; and I have seen some that in that position have extended their arms almost to the bottom, which must have been above 10 inches.

The Natural History of the Rhinoceros. By Dr. PARSons.

WHEN the rhinoceros arrived in England in 1739, Dr. Douglas went frequently to see it, for the purpose of correcting the opinions respecting it; and on June 24. of this year, exhibited before the Royal Society a drawing, with a collection of figures of that creature, taken from several authors, who

had written of him before. He mentioned 'also his dimensions; and on the 28th of the same month, he produced a collection of horns, with some account of them, but proceeded no further. Therefore, as another occasion might not offer in many years, Dr. P. gives the following account of the male rhinoceros shown in Eagle Street, near Red Lion Square, in 1739, and the drawings annexed to it. In this account, he had no regard to those of other authors, but barely described him as he often saw him, both then and afterwards.

He was fed here with rice, sugar, and hay: of the first he ate seven pounds mixed with three of sugar every day, divided into three meals; and about a truss of hay in a week, besides greens of different kinds, which were often brought to him, and of which he seemed fonder than of his dry victuals; and drank large quantities of water at a time, being then, it seems, two years old.

He appeared very peaceable in his temper, suffering himself to be handled in any part of his body; but outrageous when struck or hungry, and pacified in either case only by victuals. In his outrage he jumps about, and springs to an incredible height, driving his head against the walls of the place with great fury and quickness, notwithstanding his lumpish aspect: this Dr. P. saw several times, especially in a morning, before his rice and sugar were given to him.

In height he did not exceed a young heifer, but was very broad and thick. His head, in proportion, is very large, having the hinder part, next his ears, extremely high, in proportion to the rest of his face, which is flat, and sinks down suddenly forward towards the middle, rising again to the horn, out in a less degree. The horn stands on the nose of the animal, as on a hill. The part of the bone on which the hora is fixed rises into a blunt cone, to answer to a cavity in the basis of the horn, which is very hard and solid, having no manner of hollow nor core, like those of other quadrupeds.

That of this animal, being young, does not rise from its rough base above an inch high, is black and smooth at the top, like those of the ox-kind, but rugged downwards; the determination of its growth is backwards, instead of straight up; which is apparent, as well in the different horns of old rhinoceroses as in this of our present subject; for the distance from the base to the apex of this, backward, is not within a third part so long as that before, and it has a curved direction; and, considering the proportion of this animal's size to its horn, we may justly imagine, that the creature which bore any one of those great ones must have been a stupen

dous animal in size and strength; and, indeed, it were no wonder, if such were untractable at any rate.

If we look at him in a fore view, the whole nose, from the top of the horn to the bottom of his lower lip, seems shaped like a bell, viz. small and narrow at top, with a broad base. His under lip is like that of an ox, but the upper more like that of a horse; using it, as that creature does, to gather the hay from the rack, or grass from the ground; with this difference, that the rhinoceros has a power of stretching it out above six inches, to a point, and doubling it round a stick, or one's finger, holding it fast; so that, as to that action, it is not unlike the proboscis of an elephant.

His neck is very short, being that part which lies between the back edge of the jaw and the plica of the shoulder; on this part there are two distinct folds, which go quite round it, only the fore one is broken underneath, and has a hollow flap hanging from it, so deep that it would contain a man's fist shut, the concave side being forward. From the middle of the hinder one of these folds or plicæ arises another, which, passing backwards along the neck, is lost before it reaches that which surrounds the fore part of the body. His shoulders are very thick and heavy, and have each another fold downward, that crosses the fore leg; and, almost meeting that of the fore part of the body just mentioned, they both double under the belly close behind the fore leg.

In some quadrupeds, the fetlock bends or yields to the