any thing attained in other paths of philosophical research. But we have not vet scen all the capabilities of this wonderful power of nature. Let us therefore still follow our excellent leader in a new path of investigation." will most certainly lead to error. It is only by following his example that we can hope for his success. system. "It is surely another great recommendation of this branch of natural philosophy, that it is so simple. One single agent, a force In this spirit our author pursues his decreasing as the square of the distance inresearches into the figure of the earth, creases, is, of itself, adequate to the proand changes in the heavens, which af- duction of all the movements of the solar ford him frequent opportunities of dis- do not pass through the centre of gravity, If the direction of the projection playing those pious and religious sentithe body will not only describe an ellipse ments which are the true glory of the round the central body, but will also turn philosopher; and, after a very elegant round its axis. By this rotation, the body inquiry into the nature of the tides and will alter its form. Bat the same power enthe causes of their apparent irregulariables it to assume a new form, which is tics, he breaks out into the following animated reЯections: "726. With this we conclude our acBount of physical astronomy, a department of natural philosophy which should ever be cherished with peculiar affection by all who think well of human nature. There is none in which the access to well-founded knowledge seems so effectually barred against us, and yet there is none in which we have made such unquestionable progress; none in which we have acquired knowledge so uncontrovertibly supported, or so complete. How much, therefore, are we indebted to the man who laid the magnificent scene open to our view, and who gave us the optics by which we can examine its most extensive, and its most minute parts! For Newton not only taught us all that we know of the celestial mechanism, but also gave us the mathematics, without which it would have remained un Tu Pater et rerum Inventor. Tu patria nobis Suppeditas præcepta, tuisque ex inclyte chartis Floriferis ut apes in saltibus omnia libant, Omnia nos itidem depascimur aurea dicta Aurea, perpetuâ semper dignissima vitâ.' LUCRETIUS. For surely, the lessons are precious by which we are taught a system of doctrine which cannot be shaken, or share that fluctuation which has attached to all other speculations of curious man. But this cannot fail us, because it is nothing but a well-ordered narration of facts, presenting the events of nature to us in a way that at once points out their subordination, and most of their relations. While the magnificence of the objects commands respect, and perhaps raises our opinion of the excellence of human reason as high as is justifiable, we should ever keep in mind that Newton's success was owing to the modesty of his procedure. He peremptorily resisted all disposition to speculate beyond the province of human intellect, conscious that all attainable science consisted in carefully ascertaining nature's own laws, and that every attempt to explain an ultimate law of nature by assigning its cause is absurd in itself, and against the acknowledged laws of judgment per fectly symmetrical, and is permanent. This new form, however, in consequence of the universality of gravitation, induces a new motion in the body, by which the position of the axis is slowly changed, and the whole host of heaven appears to the inhabitants of this earth to change its motions. Lastly, if the revolving planet have a covering of fluid matter, this fluid is thrown into certain regular undulations, which are produced and modified by the same power. "Thus we see that, by following this simple fact of gravitation of every particle of matter to every other particle, through all its complications, we find an explanation of almost every phenomenon of the solar system that has engaged the attention of the philosopher, and that nothing more is needed for the explanation. Till we were put on this track of investigation, these different movements were solitary facts; and, being so extremely unlike, the wit of man would certainly have attempted to explain them by causes equally dissimilar. The happy detection of this simple and easily observed principle, by a genius qualified for following it into its various consequences, has freed us from numberless errors, into which we must have continually run while pertinaciously proceeding in an improper path. But this detection has not merely saved us from errors, but, which is most remarkable, it has brought into view many circumstances in the phenomena themselves, many peculiarities of motion, which would never have been observed by us, had we not gotten this monitor, pointing out to us where to look for peculiarities. We should never have been able to predict, with such wonderful precision, the complicated motions of some of the planets, had we not had this key to all the equations by which every deviation from regular elliptical motion is expressed. "On all these accounts, physical astronomy, or the mechanism of the celestial motions, is a beautiful department of science. I do not know any body of doctrine so comprehensive, and yet so exceedingly simple; and this consideration made nie the more readily accede to those reasons of scientific propriety which point it out as the first article of a course of mechanical philosophy. Its simplicity makes it easy, and the exquisite agreement with observation, makes it a fine example of the truth and competency of our dynamical doctrines.” We could with great pleasure transcribe more from this valuable author, whose manner of treating his subject, whether as a mathematician or a man of piety, does equal honour to his head and his heart. From the authors referred to in this work, which are very numerous, the student will know to whom to refer for the best and most copious explanation of every subject, and in every research he will be greatly assisted by the previous information he has derived from this work; which will be, we doubt not, as we said before, a standard book for the teachers of the Newtonian philosophy. ART. VI. The experienced Millwright, or a Treatise on the Construction of some of the most Useful Machines, with the latest Improvements, to which is prefixed a bort drenunt of the General Principles of the Mechanical Powers, illustrated wih Forty four Es gravings. By ANDREW GRAY, Millwright, Edinburgh. rived in one branch of the art, attempts to make a complicated machine in which other branches are concerned, is equally liable to expose himself to ridicule. Just theory and long experience form the true mechanic. Alterius sic altera poscit opem res et conjerat amice. It is evident that the writer must be cf the same opinion, or he would not have taken so much pains to communicate iaformation to his colleagues; and we should have been more pleased if, to the description of each machine, he had subjoined the best estimate he could of its powers. Millwrights are seldom corversant with this branch of their business. They are content to form a machine upon the model given to them, and their powers are chiefly employed in constructing and adjusting the parts one to another, rather than ascertaining the precise effects of the arrangement. It is with mills as with ship-building, the artists can seldom tell what is to be the effect of their work. We recommend, however, to every millwright who is desirous of improving himscit in his business, to read attentively what is said on the mechanical powers in this work, and to study the plans, by which he cannot fail of enlarging his knowledge. THE theory of the mechanical powers, well drawn up, with some remarks on motion, practical directions for the construction of machinery, the velocity of machines, and the powers of various water-wheels, precede the description of various machines, chiefly for grinding, in the construction of which the author has been concerned, and in giving their plans he has taken uncommon pains. It is justly observed by him, that "machines or engines seldom owe their origin or improvement to considerations deduced from the laws of motion. They are derived from other sources. It is from long experience and repeated trials,errors, deliberations corrections, &c. continued throughout the lives of individuals, and by successive generations of them, that the practical sciences derive their gradual advancement from awkward beginnings to their most perfect state of excellence. To be a good mechanic requires the labour of a whole life. It is an art rather perfected by practice than theory. The principles of mechanism may be learned by books, but the art must be acquired by experience." It is true indeed that a mere knowledge from books would seldom enable a man to construct a complicated machine; but it is equally true that the practical mechanic, who, from the knowledge he has deART. VII. A Dissertation on the Influence of Gravitation, considered as a Mechanic Power, explaining the Reason why the Effective Power of the same Quantity of Matter in de scending the same Height, is twice as great in its un form Descent as in its accelerated Fall, and why twice the Quantity of Resistance is required to bring a Pendulum to Rest, when gradually applied to it, as when applied at once in its lowest Point. By A. CUMMING, F. R. S. &c. THIS dissertation appeared at first to us rather too prolix; but when we considered the case of many ingenious persons with which the author must have had frequent opportunities of being acquainted, we could not blame the pains taken to place the subject in the clearest light possible. The delusion of a perpetual motion scizes the imagiration of the mechanic, and he wastes his time in a pursuit which ends in the ruin of himself and family. To show him the impossibility of obtaining his object is a great benefit conferred upon him, and in this work the idea by which many have been deluded is developed, and the nature of the effective power of gravity is examined. The more experienced mathematician will comprehend the whole in a very short time. Falling bodies, actuated merely by the force of gravity, go through about sixteen feet in the first second, and sixty-four feet in the two first seconds. The force of gravity is the same in each second, and therefore the effect of that force in the second second is, that the body should fall through sixteen feet as in the first second. Hence, by the force of gravity acting constantly on the body during the two first seconds, it is made to fall through thirty-two feet. But experience shews us that, in the two first seconds, the body falls through sixty-four feet, therefore a space of thirty-two feet is described for which we are to account. This we do by considering the velocity acquired at the end of the fall through sixteen feet, and with which, if the operation of gravity had been suspended, the body would have moved uniformly forward for ever in a right line; and the space described in the second second by this velocity being thirty-two feet, the velocity acquired at the end of the fall down sixteen feet, is such as will carry a body uniformly through twice the space in the same time. Hence, at the end of the second second, if the force of gravity were suspended, the body would move uniformly forward with twice the velocity acquired at the end of the first second; for the force of gravity communicates an additional velocity at the end of the second second, equal to that at the end of the first second. In the third second therefore, the body moves by the communicated velocity through four times the space it fell in the first second, and by the force of gravity, through a space equal to that it fell through in the first second; that is, through five times the space it fell through in the first second. In the fourth second, by parity of reasoning, it must go through seven times the space fallen through in the first second; and therefore the spaces fallen through from a state of rest, in any number of seconds, must be as the squares of the times. The last acquired velocity will also be as the times, or as the square-root of the heights. Let a body fall through a given space, which we will divide into four equal parts. The momentum of this body at the end of the fall, is as the quantity of matter multiplied into the last acquired velocity, that is, as the quantity of matter multiplied into the root of its height. The same body being checked at its fall through each of the four divisions, so as to lose its whole velocity by the check, would have a momentum at each check, varying as the quantity of matter multiplied into the root of the fourth part of the height down which the body fell without any check. That is, the momen tum of the unchecked body will be double of the momentum of the checked body at each check, but only one half of the sum of the moment at the four checks. Hence, a body falling freely by the force of gravity, will communicate only one half of the motion that the same body would do by falling through the same space divided into four equal parts, and communicating the whole of its motion at the end of each check. This being the case, it might be supposed that, by increasing the number of checks, a greater quantity of motion might be produced, and, by the application of this motion, a body that had fallen through a number of spaces might be raised again to the same height, and thus a perpetual motion be produced. But however the space down which a body falls from rest is divided, the quantity of motion, says our author, communicated by all the checks, cannot be greater than twice the momentum of the body at the end of the fall freely through the whole height. At this position we feel staggered, for upon the same principle that the author proves the sum of the moments of a body receiving the four checks in its fall, to be double the momentum of a body falling freely through that space, by dividing each of the spaces the checked body fell through, into four equal parts, and letting the body receive a check at each new division, the momentum of the body at the end of each division is one half of the momentum of the body falling through one quarter of the whole space. And the sum of the momentums of the body checked in its fall through the sixteen divisions, will be double the sum of the moments in the body checked by four divisions, that is, four times the moment of the body falling freely through the first given space. And if we divide the given space into n equal parts, at each of which a body falling receives a check, the velocity at each check being to the velocity of a body falling freely through the whole space, as unity to the root of n, the moment of the body at each check will equal the moment of the body falling freely, divided by the root of n. As there are checks, n times the moment at each check is equal to the sum of the moments of the checked body, during its fall, that is, n times the moment of the body falling freely divided by the root of n, or to the moment of the body falling freely multiplied by the root of n. Hence, if a body received one hundred checks, the sum of its moments lost would be ten times the moment of a body falling freely, at the end of its fall, through the same space, and by increasing the number of checks the proportion between the sum of the moments lost by the checked body and the moment of the body falling freely may be increased without end. But if the sum of the moments of the checked body is so much greater than that of the body falling freely, the time in which these first moments are produced is to be considered, and that in the last instance will be ten times greater in the checked body than in the body falling freely. To consider, therefore, the relation between the moments of a checked body and a body falling freely, we should suppose them to be in action exactly the same time; and now, if we suppose a body falling freely whilst another receives a hundred checks at equal distances in its fall, the body falling freely will move through ten thousand of the spaces between two adjoining checks, and its lost velocity will be ten times the velocity acquired in falling freely through the space between two adjoining checks. Hence the quantity of motion communicated by two equal bodies, the one falling freely, and the other receiving any number of checks in its fall, is the same, provided the two bodies employ the same time in their fall. We submit this examination of the question to the consideration of the author, whose language, at the conclusion of this part of his subject, we do not exactly comprehend. "After the time of the descent is prolonged, he says, to twice the time in which the body would fall the same height, no farther increase of effective power can be gained by diminishing the velocity or prolonging the time. The solicitations of gravity, after this become non-effective." This latter sentence is explained in a note, by saying that "the impulse of gravity must be obeyed with a certain degree of alacrity, otherwise it becomes non-effective." A metaphor is a very dangerous thing, in a subject very abstruse, and reduced to the precision of mathematical reasoning. The ingenious writer seems to us to have been led into the mistake, by taking the ratio of two to one, as his limit from dividing the space fallen through into four divisions only, when that ratio certainly holds; that is, the momentum of a body falling freely, is to the sum of the moments of a body checked four times, as one is to two, or as half the height fallen through to the whole height. Hence he concludes, too generally, that the quantity of motion impressed at the checks being as the number of checks or spaces, it is as the whole height, whereas the latter motion is only as half the height. But the true ratio is to be derived from the last acquired velocity of the body falling freely through the whole space, and the acquired velocity at the end of each check. The ratio of I these velocities being one to √ n is nei. ther a constant ratio nor a limited ratio. According to our position, therefore, the ratio of the sum of the moments to the moment of a body falling freely, may be increased without end; but then the time must be increased, and the moment at each check is also diminished without end; and the searcher after the perpe tual motion derives no advantage by his checks. To make this clearer, let a body of one pound weight fall fredly down ten thousand feet, its last acquired velocity will be such as to carry it uniformly through eight hundred feet in a second, and the time of fall is twentyfive seconds. Let another body of one pound fall down the same space, but receive a hundred checks in its fall, the space between each check being one hundred feet. The velocity, therefore, acquired in the fall from check to check is such as would carry the body un formly through eighty feet in a second, and the time of fall from check to check is two seconds and a half. The moment of the body when it receives a check, is to the moment of the body at the end of fall down ten thousand feet, as eighty eight hundred. But since the checked receives a hundred checks, the sum moments lost will be a hundred hty, or eight thousand. Conen the sum of the moments lost checked body is to the moment the unchecked body, as eight thoud to eight hundred, or as ten to one. The time of fall of the checked body is a hundred times two seconds and a half, that is two hundred and fifty seconds. Consequently the time of fall of the checked body is to the time of fall of the unchecked body, as two hundred and fifty is to twenty-five, or as ten to one. That is, the sum of the moments, and the time in which they are produced, are ten times greater than the moment of a body freely down the same space without checks, and the time of this fall. Our limits will not permit us to pursue our author's examination of the experiments made by Smeaton, on undershot and overshot wheels, resistance to pendulums, impracticability of forming a self-moving machine on mechanical principles by the influence of gravita tion, and other points which deserve the consideration of the mechanic: and in differing in opinion with the author we trust that he will receive our remarks as a test of the true respect which we bear to him, and which is best manifested by the evident degree of attention paid by us to his mode of reasoning. ART. VIII. Elements of Geometry; containing the first six Books of Euclid, with a Supplement on the Quadrature of the Circle, and the Geometry of Solids to which are added Elements of Plane and Spherical Trigonometry. 2d edit. enlarged. By J. PLAYFAIR, F. R. S. Sc. WHEN a rich repast is set before us, to contend on the names of the dishes instead of enjoying the liberality of our host, might seem to be an idle waste of time; and whatever title professor Playfair may choose to give to his work, we cannot doubt that his talents will render it worthy of a perusal by every mathematician. But there is a strict propriety to which all men are bound to submit; and if men in common life, or those who cultivate works of imagination and most of the sciences as they are called, may occasionally depart from it, the deviation is less pardonable in a mathematician. This work is said to contain the first six books of Euclid, by which the common reader would naturally conclude that he should here meet with six books translated from the Greek of Euclid; and if some passages differed from others in various editions, he would naturally consider the change to be due to the superior talents and investigation of the professor. This is by no means the case in the work before us, which differs from other editions of Euclid not only from a difference in the translation or emendation of certain parts, but from an entire expunction of certain passages, and introducing in their stead what appeared to the professor more valuable than the words of Euclid. Thus, the very first line of the book is not Euclid's, but the professor's. "A point is that which has position but not ANN. RAY. VOL. III. magnitude." The twelfth axiom is exchanged for this. "Two straight lines which intersect one another cannot be both parallel to the same straight line." These are alterations which cannot be allowed to an editor of a work, and much less can it be justified to change the form of demonstration of a whole book. Thus, in the fifth book, scarcely a trace of Euclid is to be seen. There it are his propositions, but the figures have disappeared, and an algebraical mode of demonstration is adopted. That there is a great advant ge in using the algebraical mode we do not deny, but then should be in union with and not to su persede that adopted by Euclid, which possesses a particular degree of ele. gance, and when well studied leaves the learner completely instructed in the doctrine of proportion, and without that study many, we fear, arrive at some degree of eminence in the mathematical world, without clear ideas on the subject. It is not necessary to point out farther deviations from Euclid. We have suf ficiently proved that this work does not contain the six books of Euclid; and it should rather be entitled Playfair's Ele-' ments of Geometry, formed upon the model of Euclid, and adopting most of the propositions, with their demonstrations, of the first six books of Euclid, How far the emendations are an improvement, may admit discussion; for 3 K |