PAPPUS

, a very eminent Greek mathematician of Alexandria towards the latter part of the 4th century, particularly mentioned by Suidas, who says he flourished under the emperor Theodcsius the Great, who reigned from the year 379 to 395 of Christ. His writings shew him to have been a consummate mathematician. Many of his works are lost, or at least have not yet been discovered. Suidas mentions several of his works, as also Vossius de Scientiis Mathematicis. The principal of these are, his Mathematical Collections, in 8 books, the first and part of the second being lost. He wrote also a Commentary upon Ptolomy's Almagest; an Universal Chorography; A Description of the Rivers of Libya; A Treatise of Military Engines; Commentaries upon Aristarchus of Samos, concerning the Magnitude and Distance of the Sun and Moon; &c. Of these, there have been published, The Mathematical Collections, in a Latin translation, with a large Commentary, by Commandine, in folio, 1588; and a second edition of the same in 1660. In 1644, Mersenne exhibited a kind of abridgment of them in his Synopsis Mathematica, in 4to: but this contains only such propositions as could be understood without figures. In 1655, Meibomius gave some of the Lemmata of the 7th book, in his Dialogue upon Proportions. In 1688, Dr. Wallis printed the last 12 propositions of the 2d book, at the end of his Aristarchus Samius. In 1703, Dr. David Gregory gave part of the preface of the 7th book, in the Prolegomena to his Euclid. And in 1706, Dr. Halley gave that Preface entire, in the beginning of his Apollonius.

As the contents of the principal work, the Mathematical Collections, are exceedingly curious, and no account of them having ever appeared in English, I shall here give a very brief analysis of those books, extracted from my notes upon this author.

Of the Third Book—The subjects of the third book consist chiesly of three principal problems; for the solution of which, a great many other problems are resolved, and theorems demonstrated. The first of these three problems is, To find Two Mean Proportionals between two given lines—The 2d problem is, To sind, what are called, three Medi<*>tates in a semicircle; where, by a Medietas is meant a set of three lines in continued proportion, whether arithmetical, or geometrical, or harmonical; so that to find three medietates, is to find an arithmetical, a geometrical, and an harmonical set of three terms each. And the third problem is, From some points in the base of a triangle, to draw two lines to meet in a point within the triangle, so that their sum shall be greater than the sum of the other two sides which are without them. A great many curious properties are premised to each of these problems; then their solutions are given according to the methods of several ancient mathematicians, with an historical account of them, and his own demonstrations; and lastly, their applications to various matters of great importance. In his historical anecdotes, many curious things are preserved concerning mathematicians that were ancient| even in his time, which we should otherwise have known nothing at all about.

In order to the solution of the first of the three problems above mentioned, he begins by premising four general theorems concerning proportions. Then follows a dissertation on the nature and division of problems by the Ancients, into Plane, Solid, and Linear, with examples of them, taken out of the writings of Eratofthenes, Philo, and Hero. A solution is then given to the problem concerning two mean proportionals, by four different ways, namely according to Eratosthenes, Nicomedes, Hero, and after a way of his own, in which he not only doubles the cube, but also finds another cube in any proportion whatever to a given cube.

For the solution of the second problem, he lays down very curious definitions and properties of medietates of all sorts, and shews how to find them all in a great variety of cases, both as to what the Ancients had done in them, and what was done by others whom he calls the Moderns. Medietas seems to have been a general term invented to express three lines, having either an arithmetical, or a geometrical, or an harmonical relation; for the words proportion (or ratio), and analogy (or similar proportions), are restricted to a geometrical relation only. But he shews how all the medietates may be expressed by analogies.

The solution of the 3d problem leads Pappus out into the consideration of a number of admirable and seemingly paradoxical problems, concerning the inflecting of lines to a point within triangles, quadrangles, and other figures, the sum of which shall exceed the sum of the surrounding exterior lines.

Finally, a number of other problems are added, concerning the inscription of all the regular bodies within a sphere. The whole being effected in a very general and pure mathematical way; making all together 58 propositions, viz, 44 problems and 14 theorems.

Of the 4th Book of Pappus.— In the 4th book are first premised a number of theorems relating to triangles, parallelograms, circles, with lines in and about circles, and the tangencies of various circles: all preparatory to this curious and general problem, viz, relative to an infinite series of circles inscribed in the space, called arbelon, arbelon, contained between the circumferences of two circles touching inwardly. Where it is shewn, that if the infinite series of circles be inscribed in the manner of this first figure, where three semicircles are described on the lines PR, PQ, QR, and the perpendiculars Aa Bb, Cc, &c, let fall from the centres of the series of inscribed circles; then the property of these perpendiculars is this, viz, that the first perpendicular Aa is equal to the diameter or double the radius of the circle A; the second perpendicular Bb equal to double the diameter or 4 times the radius of the second circle B; the third perpendicular Cc equal to 3 times the diameter or 6 times the radius of the third circle C; and so on, the series of perpendiculars being to the series of the diameters, as 1, 2, 3, 4, &c, to 1, or to the series of radii, as 2, 4, 6, 8, &c, to 1.

But if the several small circles be inscribed in the manner of this second circle, the first circle of the series touching the part of the line QR; then the series of perpendiculars Aa, Bb, Cc, &c, will be 1, 3, 5, 7, &c, times the radii of the circles A, B, C, D, &c; viz, according to the series of odd numbers; the former proceeding by the series of even numbers.

He next treats of the Helix, or Spiral, proposed by Conon, and resolved by Archimedes, demonstrating its principal properties: in the demonstration of some of which, he makes use of the same principles as Cavallerius did lately, adding together an infinite number of infinitely short parallelograms and cylinders, which he imagines a triangle and cone to be composed of.—He next treats of the properties of the Conchoid which Nicomedes invented for doubling the cube: applying it to the solution of certain problems concerning Inclinations, with the finding of two mean proportionals, and cubes in any proportion whatever.—Then of the tetragwnizousa, or Quadratrix, so called from its use in squaring the circle, for which purpose it was invented and employed by Dinostratus, Nicomedes, and others: the use of which however he blames, as it requires postulates equally hard to be granted, as the problem itself to be demonstrated by it.—Next he treats of Spirals, described on planes, and on the convex surfaces of various bodies.—From another problem, concerning Inclinations, he there shews, how to trisect a given angle; to describe an hyperbola, to two given asymptotes, and passing through a given point; to divide a given arc or angle in any given ratio; to cut off arcs of equal lengths from unequal circles; to take arcs and angles in any proportion, and arcs equal to right lines; with parabolic and hyperbolic loci, which last is one of the inclinations of Archimedes.

Of the 5th Book of Pappus.—This book opens with reflections on the different natures of men and brutes, the former acting by reason and demonstration, the latter by instinct, yet some of them with a certain portion of reason or foresight, as bees, in the curious structure of their cells, which he observes are of such| a form as to complete the space quite around a point, and yet require the least materials to build them, to contain the same quantity of honey. He shews that the triangle, square, and hexagon, are the only regular polygons capable of filling the whole space round a point; and remarks that the bees have chosen the fittest of these; proving afterwards, in the propositions, that of all regular figures of the same perimeter, that is of the largest capacity which has the greatest number of sides or angles, and consequently that the circle is the most capacious of all figures whatever.

And thus he finishes this curious book on Isoperimetrical figures, both plane and solid; in which many curious and important properties are strictly demonstrated, both of planes and solids, some of them being old in his time, and many new ones of his own. In fact, it seems he has here brought together into this book, all the properties relating to isoperimetrical figures then known, and their different degrees of capacity. In the last theorem of the book, he has a dissertation to shew, that there can be no more regular bodies beside the five Platonic ones, or, that only the regular triangles, squares, and pentagons, will form regular solid angles.

Of the 6th Book of Pappus.—In this book he treats of certain spherical properties, which had been either neglected, or improperly and imperfectly treated by some celebrated authors before his time.—— Such are some things in the 3d book of Theodosius's Spherics, and in his book on Days and Nights, as also some in Euclid's Phenomena. For the sake of these, he premises and intermixes many curious geometrical properties, especially of circles of the sphere, and spherical triangles. He adverts to some curious cases of variable quantities; shewing how some increase and decrease both ways to infinity; while others proceed only one way by increase or decrease, to insinity, and the other way to a certain magnitude; and others again both ways to a certain magnitude, giving a maximum and minimum.—Here are also some curious properties concerning the perspective of the circles of the sphere, and of other lines. Also the locus is determined of all the points from whence a circle may be viewed, so as to appear an ellipse, whose centre is a given point within the circle; which locus is shewn to be a semicircle passing through that point.

Of the 7th Book of Pappus.—In the introduction to this book, he describes very particularly the nature of the mathematical composition and resolution of the Ancients, distinguishing the particular process and uses of them, in the demonstration of theorems and solution of problems. He then enumerates all the analytical books of the Ancients, or those proceeding by resolution, which he does in the following order, viz, 1st, Euclid's Data, in one book: 2d, Apollonius's Section of a Ratio, 2 books: 3d, his Section of a Space, 2 books: 4th, his Tangencies, 2 books; 5th, Euclid's Porisms, 3 books: 6th, Apollonius's Inclinations, 2 books: 7th, his Plane Loci, 2 books: 8th, his Conics, 8 books: 9th, Aristæus's Solid Loci, 5 books: 10th, Euclid's Loci in Superficies, 2 books; and 11th, Eratosthenes's Medietates, 2 books. So that all the books are 31, the arguments or contents of which he exhibits, with the number of the Loci, determina- tions, and cases, &c; with a multitude of lemmas and propositions laid down and demonstrated; the whole making 238 propositions, of the most curious geometrical principles and properties, relating to those books.

Of the 8th Book of Pappus.—The 8th book is altogether on Mechanics. It opens with a general oration on the subject of mechanics; defining the science, enumerating the different kinds and branches of it, and giving an account of the chief authors and writings on it. After an account of the centre of gravity, upon which the science of mechanics so greatly depends, he shews in the first proposition, that such a point really exists in all bodies. Some of the following propositions are also concerning the properties of the centre of gravity. He next comes to the Inclined Plane, and in prop. 9, shews what power will draw a given weight up a given inclined plane, when the power is given which can draw the weight along a horizontal plane. In the 10th prop. concerning the moving a given weight with a given power, he treats of what the Ancients called a Glossocomum, which is nothing more than a series of Wheels-and-axles, in any proportions, turning each other, till we arrive at the given power. In this proposition, as well as in several other places, he refers to some books that are now lost; as Archimedes on the Balance, and the Mechanics of Hero and of Philo. Then, from prop. 11 to prop. 19, treats on various miseellaneous things, as, the organical construction of solid problems; the diminution of an architectural column; to describe an ellipse through five given points; to find the axes of an ellipse organically; to find also organically, the inclination of one plane to another, the nearest point of a sphere to a plane, the points in a spherical surface cut by lines joining certain points, and to inscribe seven hexagons in a given circle. Prop. 20, 21, 22, 23, teach how to construct and adapt the Tympani, or wheels of the Glossocomum to one another, shewing the proportions of their diameters, the number of their teeth, &c. And prop. 24 shews how to construct the spiral threads of a screw.

He comes then to the Five Mechanical Powers, by which a given weight is moved by a given power. He here proposes briefly to shew what has been said of these powers by Hero and Philo, adding also some things of his own. Their names are, the Axis-in-peritrochio, the Lever, Pulley, Wedge and Screw; and he observes, those authors shewed how they are all reduced to one principle, though their figures be very different. He then treats of each of these powers separately, giving their figures and properties, their construction and uses.

He next describes the manner of drawing very heavy weights along the ground, by the machine Chelone, which is a kind of sledge placed upon two loose rollers, and drawn forward by any power whatever, a third roller being always laid under the fore part of the Chelone, as one of the other two is quitted and left behind by the motion of the Chelone. In fact this is the same machine as has always been employed upon many occasions in moving very great weights to moderate distances.

Finally, Pappus describes the manner of raising great weights to a height by the combination of mechanic| powers, as by cranes, and other machines; illustrating this, and the rormer parts, by drawings of the machines that are described.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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PALISADES
PALLADIO (Andrew)
PALLETS
PALLIFICATION
PALM
* PAPPUS
PARABOLA
PARABOLOIDES
PARALLAX
PARALLEL
PARALLELISM