the sum of all its divisors; as, twelve, the sum of whose divisors, one, two, three, four, six, is sixteen; greater than the number itself.

QUESTIONS.

How is place divided by Newton?

Of

What is the definition of space? And what are its modes? What is distance? What is capacity? What is extension? What is figure? What is place? How is place divided by Aristotle? What kind of place does Clarke add to the definition ? What does Locke say of place? What is the definition of motion? How did the ancient philosophers consider motion? How is motion divided by different philosophers? what important science is motion the subject? How is duration defined? What is Locke's definition of duration? How is time defined by Locke and others? How is time distinguished? How is time measured? What is number? What is a unit? What is an integer? An even number? An odd number? A composite number? prime number? Commensurable, and incommensurable number? What are square numbers? What are cube numbers? What is a power? What is a perfect number? What is abundant number? What is deficient number.

A

CHAP. III.

METAPHYSICS-continued.

MATHEMATICS-GEOMETRY.

FROM the two great trunks of the vast tree of Creation planted and upheld by the Almighty power of God, matter and space shoot forth various branches of science which bear those fruits of art that nourish the comfort and well being of man in his present state of existence.

From the two modifications of space, form and numbers, result the sciences of geometry and arithemetic, comprised under the generic term, mathematics. The name mathematics is derived from the Greek word mathesis, science, discipline. Mathematics is that science which treats of the rates and comparison of quantities. It is defined by some, the "science of quantities," but by others more accurately, the "science of ratios, since quantities themselves are not the subject of mathematical investigation, but the ratio which such quantities bear to each other.

Mathematics are naturally divided into two classes; the one comprehending what is called pure and abstract mathematics; the other, what is styled, compound and mixed mathematics.

Pure mathematics relate to magnitudes, generally, simply, and abstractedly, and are therefore founded on the elementary ideas of quantity.Mixed mathematics are certain parts of Physics, which are by their nature, capable of being submitted to mathematical investigation. To pure mathematics, therefore, belong geometry and arithmetic.

GEOMETRY.

Geometry is the science of extension, or extended things; that is, of lines, of surfaces, or of solids.

The term geometry is composed of two Greek words, gea or gee, earth; and metreo, measure; because the necessity of measuring the earth or

certain parts and portions of it, gave birth to the invention of the principles and rules of this science. Since its first invention, geometry has been so extended, and applied to such a variety of other objects, that, united with arithmetic, it is now become the general foundation of the mathematical science.

The Egyptians are supposed to have been the first inventors of geometry, and it is imagined that the annual inundations of the Nile, by sweeping away boundaries and landmarks, impelled them to turn their minds to this subject.

From Egypt the science of geometry was carried into Greece, where it was greatly cultivated and much improved by Thales, Pythagoras, Anaxagoras, and Plato. About fifty years after the time of Plato, lived Euclid, who collected together all the theorems which had been invented by his predecessors in Egypt and Greece, and arranged them in fifteen books, entitled "the Elements of Geometry." The next after Eucliá of those ancient writers on this science, whose works are preserved, is Apollonius Pergæus, and then, follows the famous Archimedes of Syracuse. During the period of European darkness, the Arabians paid great attention to the cultivation of this as well as the other sciences, and from them the mathematics were again received into various parts of Europe. Among more modern geometricians, the names of Des Cartes, Kepler, Leibnitz, Barrow, Newton, shine with peculiar lustre. The province of geometry is astonishingly extensive. To geometry may be referred astronomy, mechanics, music; all those sciences that con

template things susceptible of addition and diminution; that is all the precise and accurate sciences. Geometrical lines and figures are not only proper to represent to the imagination the relations between magnitudes, or between things susceptible of more and less, such as spaces, times, weights, motions; but they may even represent to the mind things which it can conceive by no other means.

Geometry is commonly divided into four parts. Planimetry, altimetry, longimetry, and stereometry. Planimetry is that part of geometry which considers lines and plain figures, without any regard to heights, or depths. Thus it is restrained to the mensuration of planes or surfaces.

The art of measuring the surfaces and planes of things, is performed with the squares of long measures, as square feet, square inches, square yards; that is, by squares, whose sides are an inch, a foot, a yard; so that the area, or content of any surface is said to be found, when we know how many such square inches, feet, or yards, it

contains.

Altimetry, is that part of geometry which is concerned in the measurement of altitudes, or heights, whether accessible, or inaccessible; including the doctrine and practice of measuring both perpendicular and oblique lines, whether in respect of height or depth.

Longimetry teaches to measure lengths, both accessible, such as roads; and inaccessible, such as arms of the sea.

Stereometry is that part of geometry which concerns the measuring of solid bodies: that is,

to find the solidity, or solid content of bodies, as globes, cylinders, cubes, vessels, ships, and similar bodies.

Geometry is distinguished, likewise, into theoretical and practical.

Theoretical geometry contemplates the properties of continuity, and demonstrates the truth of general propositions, called Theorems.

Practical geometry applies those speculations and theorems to particular uses in the solution of problems.

Theoretical geometry may be subdivided into elementary; which is occupied in the consideration of right lines and plain surfaces, and solids generated from them; and sublime, which is employed in the consideration of curve lines, conic sections, and bodies formed from them.

The science of geometry is founded upon certain axioms, or self-evident truths. It is introduced by definitions of the various objects which it contemplates, and the properties of which it investigates and demonstrates; such as points, lines, angles, figures, surfaces and solids. Lines are considered as straight or curved, and in their relation to one another, as inclined, parallel, or perpendicular.

Angles are considered as right, oblique, acute, obtuse, external, vertical.

Figures are considered with respect to their various boundaries, as triangles; as quadrilaterals; as multilaterals; as circles; and as solids.

DEFINITIONS AND AXIOMS OF GEOMETRY.

A solid has length, breadth and thickness.