125 BRIEF ACCOUNT OF GEOMETRY. lumns of Sothis, on which that cele: them, he came to Athens, and was one brated person had engraven the princi- day led by curiosity to visit the schools ples of geometry, and which were dis- of philosophy. There he heard of geoposed in subterranean vases. A learned metry for the first time ; and it is procuriosity induced him to travel also into bable there is a natural adaptation of cer ing aisee boundary, the circond solid ble that he was more indebted for his instantly captivated with the subject, knowledge to the Brahmins, on the and became one of the best geometers of banks of the Ganges, than to the priests his time. He also was the first that comof Egypt. On his return, finding his posed Elements of Geometry, which, native country a prey to tyranny, he set- however, have been lost, and are only tled in Italy, and there founded one of to be regretted, because we might have the most celebrated schools of antiquity. learned from them the state of the sciHe is said to have discovered that, in anyence at that period. It has been said right-angled triangle, the square on the that, notwithstanding his want of sucside opposite the right angle is equal to cess in commerce, he retained something the two squares on the sides containing of the mercantile spirit : he accepted it, and, on this account, to have sacri- money for teaching geometry, and for ficed one hundred oxen to express his this he was expelled the school of the gratitude to the muses. This, however, Pythagoreans. "This offence, we think, was incompatible with his moral princi- might have been forgiven, in consideraples, which led him to abhor the shed- tion of his misfortunes, ding of blood on any account whatever; Two geometers, Bryson and Antiphon, and besides, the moderate fortune of á appear to have lived about the time of philosopher would not admit of such an Hippocrates, and a little before Aristoexpensive proof of his piety. The appli- tle. These are only known by some anication which the Pythagoreans made of madversions of this last philosopher on geometry gave birth to several new theo their attempts to square the circle. It, ries, such as the incommensurability of appears that before this time geomecertain lines, for example, the side of a ters knew that the area of a circle was square and its diagonal, also the doc- equal to a triangle, whose base was equal trine of the regular solids, which, al- to the circumference, and perpendicular though of little use in itself, must have cqual to the radius. led to the discovery of many propositions Having briefly traced the progress of in geometry. Diogenes Laertius has at- geometry during the two first ages after tributed to Pythagoras the merit of hav- its introduction into Greece, we come ing discovered that, of all figures having now to the origin of the Platonic school, the same boundary, the circle among which may be considered as an era in plain figures, and the square among solid the history of the science. Its celebrated figures, are the most capacious: if this founder had been the disciple of a philowas so, he is the first on record that has sopher (Socrates) who set little value on treated of isoperimetrical problems. geometry; but Plato entertained a very The Pythagorean school sent forth different opinion on its utility. After the many mathematicians ; of these, Archy- examples of Thales and Pythagoras, he tas claims attention, because of his so- travelled into Egypt, to study under the lution of the problem of finding two priests. He also went into Italy to conmean proportionals; also on account of sult the famous Pythagoreans, Philolaus, his being one of the first that employed Timæus of Locris, and Archytas, and to the geometrical analysis, which he had Cyrene to hear the mathematician Theolearned from Plato, and by means of dorus. On his return to Greece, he which he made many discoveries. He made mathematics, and especially geois said to have applied geometry to me- metry, the basis of his instructions. He chanics, for which he was blamed by put an inscription over his school, forPlato; but probably it was rather for ap- bidding any one to enter that did not unplying, on the contrary, mechanics to derstand geometry; and wien quesgeometry, as he employed motion in geo- tioned concerning the probable employmetrical resolutions and constructions. ment of the Deity, he answered, that he Democritus of Abdera studied geome- geometrized continually, meaning, no try, and was a profound mathematician. doubt, that he governed the universe by From the titles of his works, it has been geometrical laws. conjectured that he was one of the prin. It does not appear that Plato composed cipal promoters of the elementary doc- any work himself on mathematics, but trine respectiug the propertics of circles he is reputed to have invented the geoand spheres, and concerning irrational metrical analysis. The theory of the numbers and solids. He treated besides couic sections originated in this school; of some of the principles of optics and some have even supposed that Plato perspective.. himselt invented it, but there does not * Hippocrates was originally a merchant, seemn to be any sufficient ground for this but having no taste for commerce, his opinion. affairs went into disorder; to repair These discoveries innst be attributed the history, I hopn the disciple of a phon 126 BRIEF ACCOUNT OF GEOMETRY. to the Platonic school in general; for it studied at Athens, under the disciples of is impossible to say with whom each ori. Plato, before he settled at Alexandria. nated. Some of advanced years fre- Pappus, in the Introduction to the sequented the school as friends of its cele- venth book of his Collections, gives him brated head, or out of respect for his doc- an excellent character, describing him trines; and others, chiefly young persons, as gentle, modest, and benign towards as disciples and pupils. Of the first class all, and more especially such as culti. were Loadamus, Archytas, and Theate- vated and improved the mathematics. tus. Laodamus was one of the first to There is an anecdote recorded of Euclid, whom Plato communicated his inethod which seems to show he was not much of of analysis, before he made it public; a courtier : Ptolemy Philadelphus having and he is said by Proclus to have pro- asked him whether there was any easier fited greatly by this instrument of disco- way of studying geometry than that comvery. Archytas was a Pythagorean of monly taught; his reply was, " There is extensive knowledge in geometry and no royal road to geometry." This celemechanics. He had a great friendship brated man composed treatises on varifor Plato, and frequently visited him at ous branches of the ancient mathematics, e of his voyages be but he is best known by his Elements, a perished by shipwreck. Theætetus was work on geometry and arithmetic, in å rich citizen of Athens, and a friend thirteen books, under which he has coland fellow-student of Plato under So- lected all the elementary truths of geocrates, and Theodorus of Cyrene, the metry which had been found before his geometer. He appears to have cultivated time. The selection and arrangement and extended the theory of the regular have been made with such judgment, solids. that, after a period of two thousand Passing over various geometers who years, and notwithstanding the great are said to have distinguished themselves, additions made to mathematical science, but of whom hardly any thing more than it is still generally allowed to be the best the names are now known, we shall only clementary work on geometry extant, mention Menæchinus and his brother Numberless treatises have been written in Dinostratus. The former extended the since the revival of learning, some with theory of conic sections, insomuch that a view to improve and others to supplant, Eratosthenes seems to have given him the work of the Greek geometer ; but in the honour of the discovery, calling them this country, at least, they have been gethe curves of plenæchmus. His two so- nerally neglected and forgotten, and Eu. lutions of the problem of two mean pro- clid maintains his place in our schools. À portionals are a proof of his geometrical Of Euclid's Elements the first four skill. Several discoveries have been given books treat of the properties of plane to Dinostratus ; but he is chiefly known figures; the fifth contains the theory of by a property which he discovered of the proportion, and the sixth its application quadratrix, a curve supposed to have to plane figures; the seventh, eighth, been invented by Hippias of Elis. ninth, and tenth, relate to arithmetic, The progress of geometry among the and the doctrine of incommensurables Peripatetics was not so brilliant as it liad the eleventh and twelfth contain the elebeen in the school of Plato, but the sci. ments of the geometry of solids; and the ence was by no means neglected. The thirteenth treats of the five regular sosuccessor of Aristotle composed several lids, or Platonic bodies, so called because works relating to mathematics, and par- they were studied in that celebrated ticularly a complete history of these sci- school : two books more, viz. the fourences down to his own tinie: there were teenth and fifteenth, on regular solids, four books on the history of geometry, have been attributed to Euclid,.but these six on that of astronomy, and one on that rather appear to have been written by of arithmetic. What a treasure this Hypsicles of Alexandria. would have been, had we now pos- Besides the Elements, the only other sessed it! entire geometrical work of Euclid, that The next remarkable epoch in the his- has come down to the present times, is tory of geometry, after the time of Plato, his Data. This is the first in order of was the establishment of the school of the books written by the ancient geomeAlexandria, by Ptolemy Lagus, about 300 ters to facilitate the method of resoluyears before the Christian era. This tion or analysis. Iu general, a thig is event was highly propitious to learning said to be given, which is actually exhiin general, and particularly to every bited, or can be found ; and the proposibranch of mathematics then known; for tions in the book of Euclid's Data show the whole was then cultivated with as what things can be found from those much attention as had been bestowed on which by hypothesis are already kuown, geometry alone in the Platonic school. In the order of time, Archimedes is It was here that the celebrated geometer, the next of the ancient geometers that Euclid, flourished under the first of the has drawu the attention of the moderns. * Ptolemies : his native place is not cer- He was born at Syracuse, about the year tainly known, but he appears 10 have 287 A.C. He cultivated all the parts of on which hy hy BRIEF ACCOUNT OF GEOMETRY. 127 mathematics, and in particular geome. Latio versions of Commandine and Grehello try. The most difficult part of the sci- gory, and the “Data,” are among the ence is that which relates to the areas of most valuable. “ Archimedes ;" the curve lines and to curve surfaces. Archi- best edition, of which are Torelli's, in medes applied his fine genius to the snb- Greek and Latin, Oxford, 1792; and ject, and he laid the foundation of all the Peyrard's French translation, Paris, 1808. subsequent discoveries relating to it. His The first edition of the Greek test was writings on geometry are pumerous. We that of Venatorius, in 1544. Apollonius; have, iu the first place, two books on the all the writings that have been recovered sphere and cylinder; these contain the of this celebrated geometer are-l. “The beautiful discovery, that the sphere is Section of a Ratio;" and, 2. “ The Sectwo-thirds of the circumscribing cylin- tion of a Space," which were restored der, whether we compare their surfaces by Snellius, 1607; and by Dr. Halley, in or their solidities. observing that the 1706. 3. “ Determinate Section :" Sneltwo ends of the cylinder are considered lius restored these in his “ Apollonius as forming a part of its surface. He like Batavus," 1601. There is an English Wise slows that the curve surface of any translation by Lawson, to which is addsegment of the cylinder, between two ed a new restoration, by Wales, 1772. planes perpendicular to its axis, is equal Simson has restored this work in his to the curve surface of the corresponding “Opera Reliqua, 1776; and Giannini, segment of the sphere. Archiniedes was an Italian geometer, in 1773. 4.“ Tan. so much pleased with these discoveries, gencies;" Vieta restored this in his that he requested, after his death, that his * Apollonius Gallus,”1600. Some addi. tomb might be inscribed with a sphere tions were made by Ghetaldus, aad others and cylinder. by Alexander Anderson, in 1612. The Eratosthenes was another great geoine labours of Vieta and Ghetaldus have trician, and flourished in the Alexandrian been given in English by Lawson, 1771. school, about the time of Archimedes. 5. “ The Plani Loci;" these have been He was born 276 A. C., and, as a geo- restored by Schooten, 1656; and Fermeter, ranks with Aristæus, Euclid, and mat, 1679; but the best restoration is Apollonius. that of Dr. Simson, 1749. 6. “ The InAbout the time that Archimedes finish- clinations;” these were restored by Gheed his career, another great geometri taldus, in his “ Apollonius Redivivus, can appeared, named Apollonius of Perga, 1607 : to these there is a Supplement by born 240 A.C. He flourished principally Anderson, 1612; a restoration by Dr. under Ptolemy Philopater, or towards Horsley, 1770; and another by Reuben the end of that century. He studied in Burrow, 1779. Theodosius and Menethe Alexandrian school under the suc- laus, 1558, 1675, and an Oxford edition céssors of Euclid ; and so highly esteemed by Hunter in 1707. Proclus, “ CommenWere his discoveries, that he acquired the tarium in primum Euclidis Librum, liname of the Great Geometer. bri iv. Latine vertit.” F. Baroccius, The names of several other great ma- 1560. Proclus has also been ably transthematicians of antiquity, contemporary lated by Taylor, 1788. Eratosthenes's With Archimedes and Apollonius, have “Geometria,” &c. cum annot. 1672. Alcome down to us; but they are more re- bert Durer, “ Institutiones Geometricæ, terrible to a distinct work on geometry 1532. Kepler, “ Nova Steriometria,' alone, which is of too much in ULE, which is of too much importance &c. 1618. Van Culen, “ De Circulo et to be condensed into a single article of a Adscriptis," 1619. Des Cartes, “ GeoWork like this. We must, therefore, re- metrie,” 1637. Toricelli, “ Opera Geoter our readers, who would inform them- metrica,” 1644. Oughtred,'Clavis selves properly on this important guide Mathematica,” 1653. James Gregory, all excellence in art or science, to the « Geometria Pars Universalis," 1666. following works: Barrow, “ Lectiones Opticæ et GeomeOn the history of geometry, to Montu- tricæ," 1674; “ Lectiones Mathemacla, Histoire de Mathematiques," se- ticæ, 1683. David Gregory, “ Practicond edition. Bossut's “ General His- cal Geometry," 1745. Sharp, “Geome. tory of Mathematics, of which there is try Improved," &c. 1718. Stewart, å good English translation, Dr. Hut. “ Propositiones Geometricæ," 1763. ton's “ Mathematical Dictionary,” se- Thomas Simson, “ Elements of Geocond edition, 4to. Lond. 1815. Dr. metry," 1747 and 1670. “Select EserBrewster's " Edinburgh Ericyclopædia,” cises, by the same, 1752. Emerson's to which we are much indebted in this “ Elements of Geometry," 1763. Laarticle. The " Encyclopædia Metropu. croix, “ Elémens de Géometrie Descriplitana," and similar works. tive, 1795. Playfair, “ Origiu and lnOn the elements and practice of geo- vestigations of Porisms," Edin. Trans. metry, “ Euclid," of which there are vol. jii. Legendre's “ Élémens de Géomany editions; the first is that of Rat- metrie," ninth edition, 1812. Leslie, dolt, 1482. Dr. Barrow's edition of all “ Elements of Geometry, Geometrical the books, and the “ Data," and Dr. Analysis, and Plane Trigonometry,” seHorsley's of the first twelve, from the cond editiou, 1811, 128 INQUIRIES-CORRESPONDENCÉ. To such as are entering on the study of same factory, but spinning different geometry, the following works are parti. numbers of twist and weft. cularly recommended : -Simson's " Euclid," Playfair's “Geometry,” Legen. Required demonstrative proofs, dre's “ Géometrie," which is a clear and which is the best system, and whevaluable elucidation of the science, and ther any alteration is requisite or Leslie's “ Geometry.” necessary in spinning different qua lities of cotton or different numbers? · INQUIRIE S. No. 122.-CASES IN COTTON SPIN. The weights on the spinning mule NING.--A. rollers are, for single boss, the dead Take two bobbins full of stuffing weight; double boss, first, saddles out of a set, being both of them and springs, second, saddles and slipped off at the same time, and levers, with a weight hanging at the having never been broken; put them opposite end of the lever. into a roving billy, and place each Required, first, what weight ought at the roller at the same time, and there to be put upon the front and back if they continue to work, without rollers, distinctly and respectively, ever breaking, till the whole thread for the following numbers, distinis rove from the one, the other will guishing the twist and weft? Num. have at least one or two yards left, bers 30, 40, 50, 60, 70, and 80 hanks and sometimes more. twist; and Numbers 30, 40, 50, 60, A second case: There are two 70, 90, 120, and 160 hanks weft. roving billies, A and B ; each has Second, the true and correct method stuffings from the same frame; of finding the weight upon the back A is the finer roving, and has the and front rollers distinctly, in such least twist in it. Notwithstand. mules as have them weighted with ing, it has been found, on reeling springs and levers ? two caps spun on the same mule from two separate rovings, A and B, and spun at the same time, If I am so fortunate as to bring that the yarn spun from it is coarser my contemporaries into the field of than that of B. discussion on the above subjects, I Required the true cause and best shall not be the last (though, perremedy. haps, the meanest and most unwor. thy) to communicate my simple ideas; and it is my cordial wish that the subjects may be discussed in a In some spinning mules the cotton more candid and liberal manner than rollers are fixed on an equal height, the question on Long, and Short · in others the front roller is lower by Screwdrivers. For my own part, I one-eighth or one-quarter of an inch shall continue always open to conthan the other two; and I have seen viction, and ready to acknowledge some that have the foremost roller my errors and ignorance, and I hope higher at least one-eighth of an inch, to meet with reciprocal sentiments perhaps, according to the ideas of in others." the respective machinists as to the I am, Sir, plan that would answer the best for Your obedient servant,. . the cotton, or fineness and quality J. BOWKER. of the yarn. (To be continued:) Secondly,In some factories the foreman or overlooker has the fore Notices to Correspondents in our next. most top rollers fixed exactly over the centre of the bottom rollers, in others a little more forward, and in Communications (post paid) to be addressed to a third more inclined to the back ; the Editor, at the Publishers', KNIGHT and and I have seen them in all the three LACEY, 55, Paternoster-row, London. Printed by Mills, JOWETT, and Mills (late different positions in one and the BensLEY,) Bolt-court, Fleet-street. B. |