THE PRAECOGNITA GEOMETRICAL.
THe Arts, saith Arnobius, are not together with our mindes, sent out of the heavenly places, but all are found out on earth, and are in processe of time, soft and fair, forged by a continuall meditation; our poor and needy life perceiving some casual things to happen pro∣sperously, while it doth imi∣tate▪ attempt and try, while it doth slip, reform and change, hath out of these same assiduous apprehensions made up small Sciences of Art, the which afterwards, by study, are brought to some perfection.
By which we see, that Arts are found out by daily Page 2 practice, yet the practice of Art is not manifest but by spe∣culative illustration, because by speculation: Scimus ut scia∣mus, we know that we may the better know: And for this cause I first chose a speculative part, that you might the better know the practice; and therefore have first chose this speculative part of practicall Geometry, which is a Science declaring the nature, quantity, and quality of Mag∣nitude, which proceeds from the least imaginable thing.
To begin then, A Point is an indivisible, yet is the first of all dimension; it is the Philosophers Atome, such a Nothing, as that it is the very Energie of all things, In God it carryeth its extreams from eternity to eternity: in the World it is the same which Moses calls the beginning, and is his Genesis: 'tis the Clotho that gives Clio the mat∣ter to work upon, and spins it forth from terminus à quo, to terminus ad quem: in the Alphabet 'tis the Alpha, and is in the Cuspe of the Ascendant in every Science, and the house of Life in every operation. Again, a Point is either centricall or excentricall, both which are considered Geome∣trically or Optically, that is, a point, or a seeming point: a point Geometrically considered is indivisible, and being centrall is of magnitude without consideration of form, or of rotundity, with reference to Figure as a Circle, or a Globe, &c. or of ponderosity, with reference to weight, and such a point is in those Balances which hang in equili∣bra, yet have one beam longer than the other. If it be a seeming point, it is increased or diminished Optically, that is, according to the distance of the object and subject. 'Tis the birth of any thing, and indeed is to be considered as our principall significator, which being increased doth produce quantity which is the required to Magnitude; for Magnitude is no other then a continuation of Quantity, Page 3 which is either from a Line to a plain Superficies, or from a plain Superficies to a Solid Body: every of which are considered according to the quantity or form.
The quantity of a Line is length, without breadth or thicknesse, the forme either right or curved.
The quantity of a Superficies consisteth in length and breadth, without thicknesse, the form is divers, either regular or irregular; Regular are Triangles, Squares, Cir∣cles, Pentagons, Hexagons, &c.
An equilaterall Triangle consisteth of three right lines & as many angles, his inscribed side in a Circle contains 120 degrees.
A Square of four equall right lines, and as many right angles, and his inscribed side is 90 degrees.
A Pentagon consisteth of five equall lines and angles, and his inscribed side is 72 degrees of a Circle.
A Hexagon is of six equall lines and angles, and his side within a Circle is 60 degrees, which is equall to the Radius or Semidiameter.
An Angle is the meeting of two lines not in a streight concurring, but which being extended will crosse each o∣ther; but if they will never crosse, then they are parallel.
The quantity of an angle is the measure of the part of a Circle divided into 360 degrees between the open ends, and the angle it self is the Center of the Circle.
The quantity of a Solid consists of length, breadth, and thickness, the form is various, regular or irregular: The five regular or Platonick Bodies are, the Tetrahedron, Hexahedron, Octohedron, Dodecahedron, Icosahedron.
Tetrahedron is a Solid Body consisting of four equall equilaterall Triangles.
A Hexahedron is a Solid Bo∣dy consisting of six equal Squares, and is right angled every way.
An Octahedron is a Solid Body consisting of eight equal Equilaterall Triangles.
A Dodecahedron is a Solid Body consist∣ing of 12 equall Pen∣tagons.
An Icosahedron is a Solid Body consisting of 20 equal Equilaterall Triangles: All which are here described in plano, by which they are made in pasteboard: Or if you would cut them in Solid it is performed by Mr. Wells in his Art of Shadows, where also he hath fitted planes for the same Bodies.
A Parallel line is a line equidistant in all places from another line, which two lines can never meet.
Page 5A Perpendicular is a line rightly elevated to another at right angles, and is thus erected.
Suppose AB be a line, and in the point A you would erect a perpendicular: set one foot of your Compasses in A, extend the other upwards, anywhere, as at C, then keeping the foot fixed in C, remove that foot as was in A towards B, till it fall again in the line AB, then if you lay a Ruler by the feet of your Compasses, keep the foot fixed in C, and turn the other foot toward D by the side of the Ruler, and where that falls make a marke, from whence draw the line DA, which is perpendicular to AB. And so much shall suffice for the Praecognita Geometricall, the Philosophicall followeth.
