The Groundplat of the Mathematical Praeface

Here you have (as I promised) the ground plan of my Mathematical Preface, annexed to Euclid and now first published in our English tongue. An. 1570. Febr. 3.

Sciences and arts mathematical are either

Principal, which are two only,

Arithmetic.

Simple, which deals with numbers only, and demonstrates all their properties and appurtenances, where a unit is indivisible. Mixed, which, with the help of principal geometry, demonstrates some arithmetical conclusion or purpose.

Geometry.

Simple, which deals with magnitudes only, and demonstrates all their properties, affections, and appurtenances, whose point is indivisible. Mixed, which, with the help of principal arithmetic, demonstrates some geometrical purpose, as Euclid's Elements.

The use of these is either:

In supernatural, eternal, and divine things: by application, ascending. In mathematical things: without further application. In natural things, both substantial and accidental, visible and invisible, etc.: by application, descending.

The like uses and applications are found, though a degree lower, in the derivative mathematical arts.

Derivative from the principal ones: of which some have

The names of the principal ones, as:

Arithmetic, vulgar, which considers

—Arithmetic of the most usual whole numbers, and of the fractions belonging to them.—Arithmetic of proportions.—Circular arithmetic.—Arithmetic of radical numbers: simple, compound, and mixed; and of their fractions.—Arithmetic of cossic numbers, with their fractions; and the great art of algebra.

Geometry, vulgar, which teaches measuring

At hand

All lengths—Mecometrie. All planes, such as land, board, glass, etc.—Embadometrie. All solids, such as timber, stone, vessels, etc.—Stereometrie.

With distance from the thing measured, as:

How far any thing is from the measurer, seen by him on land or water, is called Apomecometrie. How high or deep any thing is from the level of the measurer's standing, seen by him on land or water, is called Hypsometrie. How broad a thing is that lies in the measurer's view, so it be situated on land or water, is called Platometrie.

From which have grown the feats and arts of

Geodesy, more skillfully to measure and survey lands, woods, waters, etc. Geography. Chorography. Hydrography. Stratarithmetrie.

Proper names, as:

Perspective—which demonstrates the modes and properties of all radiations: direct, broken, and reflected.

Astronomy—which demonstrates the distances, magnitudes, and all natural motions, appearances, and affections proper to the planets and fixed stars: for any time, past, present, and to come: in respect of a certain horizon, or without respect of any horizon.

Music—which demonstrates by reason, and teaches by sense, perfectly to judge and order the diversity of sounds, high or low.

Cosmography—which wholly and perfectly makes description of the heavenly and also the elemental part of the world: and of these parts makes homologous application and mutual collation necessary.

Astrology—which reasonably demonstrates the operations and effects of the natural beams of light and the secret influence of the planets and fixed stars in every element and elemental body: at all times, in any horizon assigned.

Statics—which demonstrates the causes of heaviness and lightness of all things: and of the motions and properties belonging to heaviness and lightness.

Anthropography—which describes the number, measure, weight, figure, situation, and color of every diverse thing contained in the perfect body of man: and gives certain knowledge of the figure, symmetry, weight, characterization, and due local motion of any designated part of the said body, and of the numbers belonging to that part.

Trochilike—which demonstrates the properties of all circular motions: simple and compound.

Helicosophy—which demonstrates the designing of all spiral lines in a plane, on a cylinder, cone, sphere, conoid, and spheroid, and their properties.

Pneumatithmie—which demonstrates by closed hollow geometric figures (regular and irregular) the strange properties (in motion or rest) of water, air, smoke, and fire in their continuity, and as they are joined to the elements next to them.

Menadrie—which demonstrates how virtue and force may be multiplied above nature's simple virtue and power: and so to direct, to lift, to pull toward, and to put or cast from, any determined virtue, weight, or force, simple or multiplied, which naturally cannot be so directed or moved.

Hypogeiodie—which demonstrates how, under the spherical surface of the earth, at any depth, to any assigned perpendicular line (whose distance from the perpendicular of the entrance, and the azimuth likewise in respect of the said entrance, being known), a certain course may be prescribed and traveled, etc.

Hydragogie—which demonstrates the possible leading of water by nature's law and by artificial help from any head (being spring, standing, or running water) to any other place assigned.

Horometry—which demonstrates how, at all times appointed, the precise customary designation of time may be known for any place assigned.

Painting—which demonstrates and teaches how the intersection of all visual pyramids made by any assigned plane (the center, distance, and lights being determined) may be represented by lines and proper colors.

Architecture—which is a science adorned with many doctrines and diverse instructions: by whose judgment all works finished by other workmen are judged.

Navigation—which demonstrates how, by the shortest good way, by the aptest direction, and in the shortest time, a sufficient ship between any two places (in a navigable passage) assigned may be conducted: and in all storms and natural disturbances happening, how to use the best possible means to recover the place first assigned.

Thaumaturgike—which gives certain order to make strange works, to be perceived by the senses, and greatly to be wondered at by men.

Archemastrie—which teaches how to bring to actual sensible experience all worthy conclusions proposed by all the mathematical arts and concluded by true natural philosophy: and both adds to them a further scope in the terms of the same arts, and also by its proper method and in peculiar terms proceeds, with the help of the aforesaid arts, to the performance of complete experiences which of no particular art can formally be challenged.

Imprinted by John Day.

An. 1570. Feb. 25.

Here you have (according to my promise) the ground plan of my mathematical preface, annexed to Euclid, now first published in our English tongue. An. 1570. Febr. 3.

Sciences and mathematical arts are either,

Principal, which are two only,

Arithmetic.

Simple, which deals with numbers only, and demonstrates all their properties and appurtenances, where a unit is indivisible. Mixed, which with the help of principal geometry demonstrates some arithmetic conclusion or purpose.

Geometry.

Simple, which deals with magnitudes only, and demonstrates all their properties, affections, and appurtenances, whose point is indivisible. Mixed, which with the help of principal arithmetic demonstrates some geometric purpose, as Euclid's Elements.

The use of which is either,

In supernatural, eternal, and divine things: by application, ascending. In mathematical things: without further application. In natural things: both substantial and accidental, visible and invisible, etc. By application, descending.

The like uses and applications are (though a degree lower) in the derived mathematical arts.

Derived from the principals; of which some have

The names of the principals, as:

Common arithmetic, which considers

—Arithmetic of the most usual whole numbers, and of the fractions belonging to them. —Arithmetic of proportions. —Circular arithmetic. —Arithmetic of radical numbers: simple, compound, mixed, and of their fractions. —Arithmetic of cossic numbers, with their fractions, and the great art of Algiebar.

Common geometry, which teaches measuring

At hand

All lengths—mecometry. All planes, as land, board, glass, etc.—embadometry. All solids, as timber, stone, vessels, etc.—stereometry.

With distance from the thing measured, as,

How far from the measurer anything is, seen by him on land or water, called apomecometry. How high or deep, from the level of the measurer's standing, anything is, seen by him on land or water, called hypsometry. How broad a thing is that lies in the measurer's view, as long as it is situated on land or water, called platometry.

From which have grown the feats and arts of

Geodesy: to measure and survey lands, woods, waters more skillfully, etc. Geography. Chorography. Hydrography. Stratarithmetrie.

Proper names, as:

Perspective,—which demonstrates the ways and properties of all radiations: direct, broken, and reflected.

Astronomy,—which demonstrates the distances, magnitudes, and all natural motions, appearances, and affections proper to the planets and fixed stars, for any time past, present, and to come, with respect to a certain horizon or without respect to any horizon.

Music,—which demonstrates by reason and teaches by sense how to judge and order perfectly the diversity of sounds, high or low.

Cosmography,—which wholly and perfectly describes the heavenly and also the elemental part of the world, and of these parts makes the homologous application and mutual comparison that are necessary.

Astrology,—which reasonably demonstrates the operations and effects of the natural beams of light and the secret influence of the planets and fixed stars in every element and elemental body, at all times, in any horizon assigned.

Statics,—which demonstrates the causes of heaviness and lightness in all things, and the motions and properties belonging to heaviness and lightness.

Anthropography, which describes the number, measure, weight, figure, situation, and color of every diverse thing contained in the perfect body of man, and gives certain knowledge of the figure, symmetry, weight, characterization, and due local motion of any part of the said body assigned, and of the numbers belonging to the said part.

Trochilike,—which demonstrates the properties of all circular motions: simple and compound.

Helicosophy,—which demonstrates the drawing of all spiral lines: in a plane, on a cylinder, cone, sphere, conoid, and spheroid, and their properties.

Pneumatithmie,—which demonstrates by close hollow geometric figures (regular and irregular) the strange properties (in motion or rest) of water, air, smoke, and fire in their continuity, and as they are joined to the elements next them.

Menadrie,—which demonstrates how, above nature's virtue and simple power, virtue and force can be multiplied; and so to direct, to lift, to pull toward, and to push or cast from, any multiplied or simple determined virtue, weight, or force that is not naturally so directable or movable.

Hypogeiodie,—which demonstrates how, under the spherical surface of the earth, at any depth, to any perpendicular line assigned (whose distance from the perpendicular of the entrance, and the azimuth likewise with respect to the said entrance, being known), a certain way may be laid out and traveled, etc.

Hydragogie,—which demonstrates the possible leading of water, by nature's law and by artificial help, from any head (being spring, standing, or running water) to any other place assigned.

Horometry,—which demonstrates how, at all appointed times, the precise usual denomination of time may be known for any place assigned.

Painting,—which demonstrates and teaches how the intersection of all visual pyramids, made by any plane assigned (the center, distance, and lights being determined), may be represented by lines and proper colors.

Architecture,—which is a science adorned with many doctrines and diverse instructions, by whose judgment all works finished by other workmen are judged.

Navigation,—which demonstrates how, by the shortest good way, by the aptest direction, and in the shortest time, a sufficient ship between any two places (in navigable passage) assigned may be conducted; and in all storms and natural disturbances happening, how to use the best possible means to recover the place first assigned.

Thaumaturgike,—which gives a set order for making strange works, to be perceived by the senses, and to make people greatly wonder.

Archemastrie,—which teaches how to bring to actual sensible experience all worthy conclusions proposed by all the mathematical arts and concluded by true natural philosophy; and it both adds to them a further scope in the terms of the same arts, and also by its proper method, in peculiar terms, proceeds with the help of the said arts to the performance of complete experiences, which cannot (formally) be challenged from any particular art.

Imprinted by John Day.

An. 1570. Feb. 25.