BRIEF ACCOUNT OF GEOMETRY. mathematics, and in particular geometry. The most difficult part of the science is that which relates to the areas of curve lines and to curve surfaces. Archimedes applied his fine genius to the subject, and he laid the foundation of all the subsequent discoveries relating to it. His writings on geometry are numerous. We have, in the first place, two books on the sphere and cylinder; these contain the beautiful discovery, that the sphere is two-thirds of the circumscribing cylinder, whether we compare their surfaces or their solidities, observing that the two ends of the cylinder are considered as forming a part of its surface. He likewise shows that the curve surface of any segment of the cylinder, between two planes perpendicular to its axis, is equal to the curve surface of the corresponding segment of the sphere. Archimedes was so much pleased with these discoveries, that he requested, after his death, that his tomb might be inscribed with a sphere and cylinder. Eratosthenes was another great geometrician, and flourished in the Alexandrian school, about the time of Archimedes. He was born 276 A. C., and, as a geometer, ranks with Aristæus, Euclid, and Apollonius. About the time that Archimedes finished his career, another great geometriclan appeared,named Apollonius of Perga, born 240 A. C. He flourished principally under Ptolemy Philopater, or towards the end of that century. He studied in the Alexandrian school under the successors of Euclid; and so highly esteemed were his discoveries, that he acquired the name of the Great Geometer. The names of several other great mathematicians of antiquity, contemporary with Archimedes and Apollonius, have come down to us; but they are more referrible to a distinct work on geometry alone, which is of too much importance to be condensed into a single article of a work like this. We must, therefore, refer our readers, who would inform themselves properly on this important guide to all excellence in art or science, to the following works: On the history of geometry, to Montucla," Histoire de Mathematiques," second edition. Bossut's General His tory of Mathematics," of which there is a good English translation. Dr. Hutton's "Mathematical Dictionary," second edition, 4to. Lond. 1815. Dr. Brewster's" Edinburgh Encyclopædia," to which we are much indebted in this article. The "Encyclopædia Metropolitana," and similar works. On the elements and practice of geometry, "Euclid," of which there are many editions; the first is that of Ratdolt, 1482. Dr. Barrow's edition of all the books, and the "Data," and Dr. Horsley's of the first twelve, from the 127 Latin versions of Commandine and Gregory, and the "Data," are among the most valuable. "Archimedes;" the best edition, of which are Torelli's, in Greek and Latin, Oxford, 1792; and Peyrard's French translation, Paris, 1808. The first edition of the Greek text was that of Venatorius, in 1544. Apollonius all the writings that have been recovered of this celebrated geometer are-1. "The Section of a Ratio;" and, 2. "The Section of a Space," which were restored by Snellius, 1607; and by Dr. Halley, in 1706. 3. Determinate Section;" Snellius restored these in his "Apollonius Batavus," 1601. There is an English translation by Lawson, to which is added a new restoration, by Wales, 1772, Simson has restored this work in his "Opera Reliqua, 1776; and Giannini, an Italian geometer, in 1773. 4. "Tangencies;" Vieta restored this in his "Apollonius Gallus," 1600. Some additions were made by Ghetaldus, and others by Alexander Anderson, in 1612. The labours of Vieta and Ghetaldus have been given in English by Lawson, 1771. 5. "The Plani Loci;" these have been restored by Schooten, 1656; and Fermat, 1679; but the best restoration is that of Dr. Simson, 1749. 6. "The Inclinations;" these were restored by Ghetaldus, in his " Apollonius Redivivus, 1607 to these there is a Supplement by Anderson, 1612; a restoration by Dr. Horsley, 1770; and another by Reuben Burrow, 1779. Theodosius and Menelaus, 1558, 1675, and an Oxford edition by Hunter in 1707. Proclus, "Commentarium in primum Euclidis Librum, libri iv. Latine vertit." F. Baroccius, 1560. Proclus has also been ably translated by Taylor, 1788. Eratosthenes's 66 Geometria," &c, cum annot. 1672. Albert Durer," Institutiones Geometricæ, 1532. Kepler, "Nova Steriometria,' &c. 1618. Van Culen, "De Circulo et Adscriptis," 1619. Des Cartes, "Geometrie," 1637. Toricelli, "Opera Geometrica," 1644. Oughtred, "Clavis Mathematica," 1653. James Gregory, "Geometria Pars Universalis," 166. Barrow, "Lectiones Opticæ et Geometrica," 1674;" Lectioues Mathematica," 1683. David Gregory, "Practical Geometry," 1745. Sharp, "Geome try Improved," &c. 1718. Stewart, 66 66 Propositiones Geometricæ," 1763. Thomas Simson, "Elements of Geometry," 1747 and 1670. "Select Exercises," by the same, 1752. Emerson's "Elements of Geometry," 1763. Lacroix, Elémens de Géometrie Descriptive," 1795. Playfair, "Origin and Investigations of Porisms," Edin. Trans. vol. iii. Legendre's "Élémens de Géometrie," ninth edition, 1812. Leslie, "Elements of Geometry, Geometrical Analysis, and Plane Trigonometry," second edition, 1811. 128 INQUIRIES CORRESPONDENCE. To such as are entering on the study of geometry, the following works are particularly recommended :-Simson's "Euclid," Playfair's "Geometry," Legen. dre's" Géometrie," which is a clear and valuable elucidation of the science, and Leslie's" Geometry." INQUIRIES. No. 122.-CASES IN COTTON SPINNING.-A. Take two bobbins full of stuffing out of a set, being both of them slipped off at the same time, and having never been broken; put them into a roving billy, and place each at the roller at the same time, and if they continue to work, without ever breaking, till the whole thread is rove from the one, the other will have at least one or two yards left, and sometimes more. A second case:-There are two roving billies, A and B; each has stuffings from the same frame; A is the finer roving, and has the least twist in it. Notwithstanding, it has been found, on reeling two caps spun on the same mule from two separate rovings, A and B, and spun at the same time, that the yarn spun from it is coarser than that of B. Required the true cause and best remedy. B. In some spinning mules the cotton rollers are fixed on an equal height, in others the front roller is lower by one-eighth or one-quarter of an inch than the other two; and I have seen some that have the foremost roller higher at least one-eighth of an inch, perhaps, according to the ideas of the respective machinist's as to the plan that would answer the best for the cotton, or fineness and quality of the yarn. Secondly,-In some factories the foreman or overlooker has the foremost top rollers fixed exactly over the centre of the bottom rollers, in others a little more forward, and in a third more inclined to the back; and I have seen them in all the three different positions in one and the same factory, but spinning different numbers of twist and weft. Required demonstrative proofs, which is the best system, and whether any alteration is requisite or necessary in spinning different qualities of cotton or different numbers? C. The weights on the spinning mule rollers are, for single boss, the dead weight; double boss, first, saddles. and springs, second, saddles and levers, with a weight hanging at the opposite end of the lever. Required, first, what weight ought there to be put upon the front and back rollers, distinctly and respectively, for the following numbers, distinguishing the twist and weft? Numbers 30, 40, 50, 60, 70, and 80 hanks, twist; and Numbers 30, 40, 50, 60, 70, 90, 120, and 160 hanks weft.Second, the true and correct method of finding the weight upon the back and front rollers distinctly, in such mules as have them weighted with springs and levers ? If I am so fortunate as to bring my contemporaries into the field of discussion on the above subjects, I shall not be the last (though, perhaps, the meanest and most unworthy) to communicate my simple ideas; and it is my cordial wish that the subjects may be discussed in a more candid and liberal manner than the question on Long and Short Screwdrivers. For my own part, I shall continue always open to conviction, and ready to acknowledge my errors and ignorance, and I hope to meet with reciprocal sentiments in others. I am, Sir, Your obedient servant, ». (To be continued.) Notices to Correspondents in our next. Communications (post paid) to be addressed to the Editor, at the Publishers', KNIGHT and LACEY, 55, Paternoster-row, London. Printed by MILLS, JOWETT, and MILLS (late. BENSLEY,) Bolt-court, Fleet-street. Mechanics' Magazine, MUSEUM, REGISTER, JOURNAL, AND GAZETTE. No. 93.] SATURDAY, JUNE 4, 1825. [Price 3d. "It is always requisite to think justly, even in matters of small importance."-Fontenelle.. 130 PERPETUAL PUMP-ON THE SLIDING RULE. PERPETUAL PUMP. SIR, Travelling in a distant country, I observed, by the roadside, a horse-trough flowing over, being fed by a spout continually running this spout was supplied by a Perpetual Pump, in a very small stream about a quarter of a mile distant, which I went to examine, and found it to be a contrivance so extremely simple in itself, and so useful in effect, that I take the opportunity of sending a description of it to you, thinking it may be worthy of a place in your valuable repository of mechanical knowledge. The figure is so simple, it scarcely requires explanation. By the rough sketch prefixed, it is clearly perceived that two or more boards are placed across the stream, and held up by five or six stakes driven into the stream perpendicularly, and nailed to them: in the upper board (not exactly in the middle of the stream) is cut a notch for a spout, through which the whole of the brook or stream is conducted, and passes over as a spout into a sort of oblong box, whose outer end is formed somewhat like a shovel; when this is full, it overbalances the stones on the other side of the pivot, and, descending, instantly empties itself, and is as instantly brought to its former level by the stones in the frame on the opposite end. This process is repeated every time the stream fills the box, the frequency of which, of course, depends upon the magnitude of the stream; in the machine I saw, it was about twice a minute. As the crate of stones rises, it lifts up the rod affixed to the pump handle, and when the water on the other side is emptied, the weight of the stones pulls down the pump handle, and so keeps constantly performing. A continuation of troughs or gutters conveyed it to the place required, where it never ceased running, fully answering its intended purpose. Two stakes are driven in the stream, for the axles of the machine to rest on, as in the figure. ON THE SLIDING RULE. SIR,-Some time since I endeavoured to explain the construction and application of Gunter's Line, and should have extended the subject to the use of the same lines, as en graven on the Sliding Rule, had I not thought that some of your practical Correspondents would have been induced to take up the subject where I left it, as I am confident that those who are in the daily use of this instrument, are much more competent to give a familiar explanation, suited to the capacities of workmen in general, than I can be. However, as I find myself called upon by your Correspondent, "Monad," and alassistance, when I have it in my ways feeling desirous of lending my power, towards the elucidation of mechanic, I shall proceed to give any subject that may benefit the such a description of the instrument that, I trust, will enable any one to estimate its utility, and apply to practice Gunter's Line, as adapted to the art of measuring by the sliding rule; and I trust that my ready acquiescence to the wishes of Monad will induce him to favour your readers with the use and application of the sectoral lines, as applied to mensuration and mechanics, as also the method of their construction. I shall now proceed with the subject, first remarking, that the term ON THE SLIDING RULE. "slide rule" is applied to a variety of instruments known under that title, such as Partridge's, Hunt's, Everard's, Coggershall's, &c.; but that most commonly in use is Coggershall's, and known by the general term of the carpenter's two-foot slide rule; and as that is the most simple, I shall give its use and application previous to that of some others, which are adapted to other purposes than performing the rules of multiplication, division, and extraction of roots. First, then, Of the Construction of the Slide Rule. This instrument consists of two pieces of box wood, each a foot in length, and connected together by a brass rule joint; one side or face of the rule is divided into twelve inches, and numbered; when the rule is open from 1 to 24, each inch is also subdivided into halves, quarters, and eighths (or half-quarters), the use of which speaks for itself; the remainder of this side of the rule is taken up in general with scales for planning dimensions. The rule I have before me contains only two scales on this face, the one an inch and a quarter scale, and the other an inch and a half; each of the large divisions, in drawing or measuring plans, are called feet, and one of these divisions in each scale is divided into twelve equal parts, corresponding to inches on the plan. The use of these scales is so obvious that it needs no illustration; but, for the sake of the young draftsman, I shall merely state that, in laying down a plan of any building or piece of work on paper, suppose he wishes to represent the distance of five feet six inches, he has nothing to do but to take from the scale a distance between the compasses equal to five of the large divisions and six of the smaller, and draw a line of that length on his paper; and if he wishes his drawing to be of such dimensions that one foot of the work required to be executed shall be represented on the paper by an inch and a half, he must use the scale marked 1, and so of all the other scales. We will next observe that the thin edge of the rule is divided into, first, 131 ten equal parts, which answer to the tenth part of a foot, and then these parts are subdivided again into ten equal parts also, which answer to the hundredth part of a foot; and thus we have the foot decimally divided, (and which, by-the-bye, we have to regret is not the general practice, as it would much shorten calculation, and be less burdensome to the memory than the present system of dividing the foot into twelve equal parts). These decimal divisions on the rule are continued along the two legs of it, and thus we have the whole length of two feet divided into 200 equal parts. We now proceed to the other face of the rule, and which it is the chief object of this paper to describe. One leg contains a continuation of the scales for planning, from one quarter of an inch to one inch to the foot: on the other leg is the division more especially under our consideration, and which I shall endeavour to describe as familiarly as possible, that their nature, being well understood, will contribute to the thorough knowledge of their use and application. On this face are four lines, marked A, B, C, D; the three uppermost of which, marked A, B, C, are exactly similar to the lines on Gunter's scale, being the logarithms (or artificial numbers) from one to ten, twice repeated (see the description of this line, page 157, vol. III. of this Magazine), and which it is, therefore, unnecessary here to describe. The first of these lines, A, is engraven on the rule itself; the two next, B and C, are engraven on a brass or wood slider, and are exactly similar to the line A; and these three lines are called double lines, as proceeding from unity to ten, and twice repeated: the fourth line, marked D, is a single logarithmetic line (similar in its properties to the other lines), and is the representation of the logarithms of the numbers, the proportional lengths of which are double those of the lines A, B, and C, thus answering to cubic measure or solid content of bodies whilst the other lines, A, B, C, answer to superficial measurement, as it is well known that the logarithm |