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Number is the Semi-difference of the Faflums of the Square of the Side into the Number of Angles, diminifhed by two Units; and of the Side itfelf into the Number of Angles di- rninifh'd by four Units.
The Sums of Polygonal Numbers collected in the fame manner as the Polygonal Numbers themfelves are out of A- rithmetical Progremons ; are called Pyramidal Numbers. See
PyRAiMIDAL.
POLYGRAPHY, Polygraphia, Polygra- ph i c e, the Art of writing in various unufual, Manners or Cyphers; as alfo of decyphering the fame. See Writing.
The Word is ufually confounded with Steganography and Cryptography. See Steganography and Cryptography.
The Antients feem to have been very little acquainted with this Art; nor is there any Marl; of their having gone beyond the Lacedemonian Scytala. See Scytala.
Trithemius, Porta,V!genere,wi Father Niceron, have wrote On the fubjeft of Polygraphy or Cyphers. See Cypher.
The Word is form'd from the Greek mn), iimltum, and y&t», fcriptura, writing.
POLYHEDRON, P o l y e d r o n, in Geometry, a Body comprehended under feveral Faces or Sides. See Bo- dy and Solid.
Such are all the five regular Bodies; viz.. the Tetrahedron, Oftahedron, Cube or Exahedron, Dodecahedron, and Ico- fihedron. See each under its proper Article.
If the Sides of the Polyhedron be regular Polygons, all fi- milar and equal ; the Polyhedron becomes a regular Body, and may be infcribed in a Circle. See Regular Body., &c.
Gnomonic Polyhedron, is a Stone with feveral Fa- ces, whereon are projected various kinds of Dials. See Dial.
Of this kind that in the Privy-Garden, London, now gone to ruin, was anciently the fineft in the World.
The Word is form'd from the Greek «iM>> much, and eJVs;, Seat.
Polyhedron, or Polyscope, in Opticks, is a Glafs or Lens confiding of feveral plain Surfaces, difpoled in- to a Convex Form ; popularly call'd a Multiplying-Glufs. See Lens and Multiplying Glafs.
The Phenomena of the Polyhedron are as follow.
Dollrine t/tfePoiYHEDROK, or Multiplying-Glafs.
If feveral Rays, as EF, A B, CD, (Tab. Optich, Fig. 71.) fall parallel on the Surface of a Polyhedron ; they will conti- nue parallel after Refraftion. See Ray and Refraction.
If then the Polyhedron be fuppofed regular ; L H, H I, I M, will be as Tangents cutting the Spherical Convex Lens in F,B and D; confequently Rays falling on the Points of
Contaft interfeft the Axis. Wherefore, fince the reft are
parallel to thefe; theyalfo will mutually interfeft eachother in G.
Hence, if the Eye be placed where the parallel Rays de- cuffite; Rays of the fame Objeft will be propagated to it ftill parallel from the feveral Sides of the Glafs. Wherefore fince the Crystalline Humour, by its Convexity, unites pa- rallel Rays; the Rays will be united in as many different Points of the Retinaj a,b,c, as the Glafs has Sides.
Confequently, the Eye, thro' a Polyhedron, fees the Ob- jeft repeated as many times as there are Sides.— And hence, fince Rays, coming from remote Objefts, are parallel ; a re- mote Objeft is Teen as often repeated thro' a Polyhedron as that has bides.
2. If Rays, AB, AC, AD, (i%. 72.) proceeding from a Radiant Point A, fall on feveral Sides of 1 regular Polyhedron ; after Refraction they will decuffate in G ; and proceed on a little diverging.
Hence, if the Eye be placed where the Rays coming from the feveral Planes, decuffate; the Rays will be propagated to it from the feveral Planes a little diverging, i. c. as if they proceeded from different Points. But fince the Cryftalline Humour by its Convexity, collefts Rays from feveral Points into the fame Point ; the Rays will be united in as many different Points of the Retina, a, b, c, as the Glafs has Sides. Confequently the Eye being placed in the Focus G, will fee even a near Objeft repeated as often thro' the Po- lyhedron as that has Sides.
Thus may the Images of Objefts be multiplied in a Ca- mera obfcura; by placing a Polydron at its Aperture, and
adding a Convex- Lens at a due Diftance therefrom. And
it really makes a very pleafant Appearance, if aPrifm be ap- plied fo as the colour'd Rays of the Sun refrafted therefrom be received on the Polyhedron. Form by this means they will be thrown on a Paper, or Wall near at hand in little lucid Specks, much exceeding the brightnefs of any preci- ous Stone ; and in the Focus of the Pslybcdron, where the Rays decuffate, (for in this Experiment they are received on the convex Side) will be a Star of furprizing Luftre.
If Images be painted in Water-Colours in the Areola or little Squares pf a Polyhedron, and the Glafs applied to the
Aperture of aCamera obfcura; the Sun's Rays parting thro" it will carry with them the Images thereof, and projcft them on the oppofite Wall.
This Artifice bears a Refemblance to that other, whereby an Image on Paper is projefted on the Camera, ■viz.. by wetting the Paper with Oil, and (training it tight on A Frame ; then applying it to the Aperture of the Camera ob- fcura, fo as the Rays of a Candle may pais through it upon the Polyhedron. See Camera.
To make an Anamorphofu or deform d Image, whichthro' a Po^ lyhedronor Multiplying Glafs fliall appear regular and beautiful.
At one End of a Horizontal Table erect another at right Angles, whereon a Figure may be delign'd ; and on the other End ereft another ; to ferve as a Fulcrum or Sup- port, moveable on the horizontal one. To the Fulcrum
apply a Piano Convex Polyhedron, confuting e. gr. of 24 plain Triangles; let the Polyhedron be fitted in a Draw Tube, whereof that End towards the Eye to have only a very fmall Aperture, and a little further off than the Focus.—. Remove the Fulcrum from the other perpendicular Table, till it be out of the Diftance of the Focus; and that more^as the Image is to be greater.— Before the little Aperture place a Lamp ; and trace the Luminous Areola; projefted from the Sides of the Polyhedrm, with a black Lead Pencil, on the vertical Plane, or a Paper apply'd thereon.
In thefe feveral Areote, defign the feveral Parts of an I- mage, in fuch manner as that when join'd together they may make one whole ; lookinga-frefh, every now and then thro' the Tube, to guide, correft, c>c. the Colours, and to fee that the feveral Parts match aptly together.
The intermediate Space fill up with any Figures or De- figns at Fleafure ; contriving it fo as that to the naked Eye the whole may exhibit fome Appearance very different from that intended to appear through the Polyhedron.
The Eye, now, looking thro' the little Aperture of the Tube, will fee the feveral Parts and Members difpers'd a- mong the Areolae to exhibit one continued Image; all the intermediate ones difappearing. See Anamorphosis.
POLYHISTORES. See History, Polyma- th y, &c.
FOLYMATHY,PoLYMATHiA,theKnowledge of ma- ny Arts and Sciences ; or an Acquaintance with a great Num- ber of different Subjefts. See Encyclopedia.
The Word comes from the Greek tnto), multum, and p»- Smw, Knowledge, Learning.
Lipfius, Scaliger, Kircher, Petavius, Politian, Salmafiusi &c. were famous for Polymathy.
Among the Ancients, fuch as were eminent this Way; were called Polyhiftores. See History.
Polymathy is frequently little more than a confufed Heap of ufelefs Knowledge occafionally detail'd, either pertinently
or impertinently, for Parade The genuine Polymathy is an
extenfive Erudition , or a Knowledge of a great Number of Things, well digefted, and applied to the Purpofe, and ne- ver but out of Necerfity.
POLYMYTHY, Polymythia, in Poetry, a Multi- plicity of Fables, in an Epic or Dramatic Poem ; in lieu of an Unity, or a fingle one. See Fable, Unity, &c.
Polymythia is a very great Fault.— It confiits in aflem- bling a Number of diftinft Aftionsor Fables into one conir plex Body. See Action.
Such a Work Boffu compares to the Batrachomymachiao J or one of the Fables of Efop : and fuch would be the Idea of a Thefeid, an Heracleid, an Achilleid, or the like Poems, which mould comprehend all the Aftions of thofe Heroes; compared with the Iliad, or /Eneid. See Hero, Epic, &c.
POLYNOMIAL, or rather Multinomial, Roots, in Ma- thematicks. See Multinomial and Root.
P O L Y P TR U M, in Opticks, a Glafs through which Objefts appear multiplied, but diminilhed. See Mdlti-
PLICATION.
The Polyoptrum differs both in Structure and Phenome- na from the common Multiplying Giaffes, call'd Polyhedra. See Polyhedron.
The Word Polyoptrum is form'd from the Greek mhl, much, many, and »:T75f«u> I fee.
Conjhuclio?! of the Polyoptrum.
In a Glafs, plain on both Sides, A B, (Tab. Opt. 2%. 73.) and about three Fingers thick, cut out fpherical Segments, fcarce a fifth Part of a Digit in Diameter.
If then the Glals be removed from the Eye, till you can: take in all the Cavities at one View, you will fee the fame Objeft as if thro' fo many feveral concave Glaflcs, as there are Cavities, and all exceedingly fmall.
Fit this, as an Objeft Glals, in a Tube A B C D, whole Aperture A B is equal to the Diameter of the Glafs, and the other C D equal to that of an Eye Glals; e. gr. ab. % ut a Finger's Breadth. The Length of the Tube A C to be ac- commodated to the Objeft and Eye-Glafs, by Trial.
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