quently 4 must be a complete nth power, or AA"; that is

which muft necessarily be an integer. But these three frac tions are each in their lowest terms, because r, s, and t, are prime to each other, and each of the denominators contains a factor that is not common to the other two; they cannot therefore be equal to an integer, by cor. Lemma 1, and consequently the equation is impossible under the first condi tion. And in order to arrive at the results of the other three conditions, we have only to substitute n"-" instead of r", - for s", and "-" for ", whence we draw the follow ing four conclusions,

according as we assume the 1st, 2nd, 3d, or 4th, condition. And in each of these expressions we ought to have A', A", A"", A"", integer numbers, if the given equation were possible; but since in each of these expressions we have three fractions each in its lowest terms, and the denominator of each contains a factor not common to the other two, therefore by Lemma 1, and its corollary, they cannot produce an integer number.

Having shown, therefore, that, if the equation 1-y”—*^ were possible, one of the quantities A', A", A", or A"""a

would

would be an integer, and having also demonstrated, that no one of these quantities can be an integer, it follows that the equation whence they were derived is impossible; that is the equation xyz" is impossible, when n is a prime number.

But we have also shown, that the impossibility of the equa tion "—y"—", when a is a prime, involves with it the impossibility of every equation of the form

x2 + y" = 2",

in which n is any number whatever except 2, or some power of 2; and we have likewise shown, that the impossibility of the equation, when n is any power of 2, is involved in that of x^—y*=z*, which particular case has been proved to be impossible by Euler; and consequently the equation

is always impossible, when is any integer number whatever greater than 2. And since the equation "+y" =2";

Method of raising large Stones out of the Earth; by Mr. Ro-
BERT RICHARDSON, of Keswick, in Cumberland*.

GENTLEMEN,

detached

I, Robert Richardson, of Keswick, in the parish of Cros- New inethed thwaite, and county of Cumberland, beg leave to inform of taking large you, that I have found out a method of taking large self- stones out of stones out of the ground in a very expeditious manner; and the ground. that by this meaus two men will take as many stones out of the ground in one day, as would require twelve men in

Trans, of the Soc. of Arts, vol. xxvi, p. 190. The silver medal was voted to Mr. Richardson for this invention.

the

the usual way of blasting, and afterward using large levers, &c.

Where stones from two to four tons each are to be taken up, two men will raise as many as twenty men in the usual way. The work is done by the power of a tackle, but by my method of fixing the tackle to the top of the stone, by the plug which I have invented, it will hold till the stone is pulled out of the ground, and laid upon the surface, or upon a carriage, if required, all which can be done in a very little time.

Stones of four tons weight, or upwards, may be taken out of the ground within the time of five or ten minutes, by two men, without any earth or soil being previously taken from around them, or without any digging with hacks or spades. J. C. Curwen, Esq., of Workington, has seen and approved of my performance with this invention, and if the Society should think it deserving of a premium, it would ever be gratefully acknowledged by

Gentlemen,

Your most obedient humble servant,
ROBERT RICHARDSON.

Keswick, Feb. 8, 1808.

DEAR SIR,

One of the plugs is

There is no diffi-.

I cannot suffer Mr. Richardson's letter to be sent to the Society, without adding a few lines concerning it. I can bear ample testimony to the ease, with which the largest self-stones are lifted by this method. I have seen one upwards of five tons lifted by four men. sent for the inspection of the Society. culty in cutting the hole to receive it, the only care is not to make it too large. It is difficult to explain the theory of its action; the least stroke laterally disengages the stone, In many situations it is likely to be of great use, not only in drawing stones out of the ground, but in making weirs and embankments, where the stones are only to be lifted a moderate height.

One of my farmers in Westmoreland has made great use of one, and speaks of it in high terms. I have exhibited it to numbers of persons, who could not believe its power, till they saw it tried.

Mr.

Mr. Richardson submits its examination to the Society, and I conceive it will be very useful and beneficial in-cases of new enclosures of land. I do not think it would answer for soft stones, or safe to use for raising stones in buildings, it being so easily disengaged by any lateral blow. By adding wheels to the tackle machine, or having it upon a sledge, a great deal of time and trouble would be avoided. I pur pose to employ this method next summer in making an em bankment against the sea; the facility it will give in raising and removing large stones will expedite the work greatly. If any farther certificates of the performance of this plug be required by the Society, I wili with pleasure transmit them to you. I will answer for its extracting any stone not ex❤ ceeding five tons weight out of the ground, without any previous moving of the earth; and it is of importance to preserve large stones entire.

I am, with respect, dear Sir,

Your obedient humble Servant,

Workington Hall, Feb. 19, 1808.

SIR,

J. C. CURWEN.

I am favoured with your letter, desiring my opinion of Farther re the utility of the iron plug invented by Robert Richardson, marks on it. of Keswick. That which I use is about six inches long, and one inch and a quarter in diameter; it requires a hole of its own size, only two inches deep; the plug is to be driven in a little short of the bottom, and will raise a stone of six or eight tons, with the assistance of three men, in the course of ten minutes after the hole is prepared; and I do not hesitate to say, that three men, thus furuished, will clear the ground of large stones in less time, and more ef fectually, than twelve men by any other method yet come to my knowledge. The plug should be made of good beaten iron. The simplicity and cheapness of the whole apparatus is a great object, as a good plug of the size I use will cost only two shillings and sixpence. I am fully of opinion, that by adding more and stronger ropes and pullies, work might be done by it to an amazing extent. I have reaped great advantage in my farm from the aid of the iron

plug,

The method described,

plug, and, in justice to the inventor, am happy in thus vouching for its extreme usefulness. Several of my respectable neighbours have experienced the aid and benefit of the above instrument, and will vouch, if required, for the truth of the above statement.

I am, Sir,

Your truly obedient Servant,
ROBERT WRIGHT,

Rose Gill Hall, near Shap, West

moreland, May 9, 1808.

Reference to the Engraving of Mr. Richardson's Invention for raising large Stones out of the Earth. See Pl. VI, Figs. 4, 5, 6, and 7.

Fig. 4, K, shows the upper part of a stone nearly buried in the earth, having a hole made in it three inches and a half deep, and one inch in diameter, by means of a miner's jumper; the cylindrical tail of the plug a, figs. 5, 6, and 7, which is of the same size, is driven fast into it, by means of a hammer applied upon the head of the plug at G. This plug, in its whole length, is nine inches, and has a hole made in its broad part H, through which the oval iron ring B passes easily, and on which the plug can move backwards and forwards, when the ring is hung upon the hook of the lower pulley block of the lifting tackle. CCCC represent the four legs or frame work of the quadrangle; D a fivefold tackle, with blocks ten inches in diameter; E a roller seven inches in diameter, turned by two long iron levers bb; the handle I is used as a safeguard, and to assist to regulate the power of the levers. In fig. 4, the plug A is shown fixed in the stone K, ready to draw it out of the ground, by means of the lifting tackle.

N.B. The hinder legs of the quadrangle are made to close in between the fore legs, for the convenience of carriage.