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(In the marginal references, the Roman numerals express the Book ; and the Arabic ones
next following, the Proposition. The Intercalary Book is expressed by INTERC.)
1. AFTER a thing, or a class of things, has been rendered obvious
to the senses, and either by speech, writing, pictorial representation, or examination in its proper substance, has been made sufficiently the subject of knowledge to be distinguished from other things; the word appointed to signify it in future, is called its name. The giving of names for the purposes of science, is called Nomenclature.
II. Anything that can be made the object of touch, is called
III. A body whose particles are immoveable among themselves,
at least by any force there is question of employing ; is called a hard body.
IV. That which has length, breadth, and thickness, is called a
or by the internal. The first may be called a substantial solid; the
is a hollow cube. If nothing else be specified, the solid is understood to be substantial.
V. That which bounds a solid, is called a surface.
A surface, consequently, has length and breadth, but not thickness.
For if it had thickness, it would not be the boundary, but part of the solid.
VI. That which bounds a surface, is called a line.
A line, consequently, has length, but not
thickness or breadth. For the surface
not be the boundary, but part of the surface.
manifestly have neither breadth nor thickness. And if to save trouble
the same surface is designated by black scrawls instead, it is in strictSee Note.
ness not the scrawl, but the edge of the scrawl, that is the line. VII. The extremity of a line, is called a point.
A point, consequently, has position, but not dimensions of any kind.
For the line itself has neither thickness nor breadth ; wherefore its extremity can have neither. And if the so-called extremity had
length, it would not be the extremity, but part of the line. Thus the extremity of a line in the black surface above, is a point ;
which manifestly has no dimensions of any kind. And if the lines are represented by black scrawls, the point is in strictness the
extremity of one of their edges. When a point is directed to be taken in some part of a line which is
not the extremity, it may be imagined to be determined by causing See Note.
the line to terminate at that point. See Note. VIII. Anything that has boundaries which are fixed, is called
solid, surface, or line. To which may hereafter be added angular,
or those of which the subject is an angle. Figures of all kinds, lines, and points, will always be considered as ex.
hibited on a hard body of some kind, which causes the position of the several parts or points to be fixed with relation to one another; and will, on occasion, be supposed to be turned about an assigned point or points, in any manner that can be shown to be practicable with
the hard body on which they are understood to be represented. Nevertheless the application of one object to another will, when required,
be imagined to take place without bar of corporeal substance ;--that is to say, without impediment from the existence of other parts than those it is desired to compare. Which, though avowedly only an act of the imagination, is sufficient for obtaining the results the geometer desires.
Things which occupy the same place, are said to coincide. The only way in which coincidence can actually be brought to pass, is
when the things touch one another or are in contact. Thus there is coincidence between the surface of a cast and the surface of the mould which contains it. And in like manner two lines or points
may actually touch one another and coincide. But besides this, there is an imaginary coincidence frequently appealed
to by geometers; which is, when it can be shown that coincidence would take place, if the objects in question could be applied to one another without bar of corporeal substance as intimated above. And in the same manner that two objects may be thus imagined to coincide, may three, or any other number.
X. Points which do not coincide, are said to be distant.
XI. Two points A and B are said to be equally distant with
two others C and D, or to be at a distance equal to the distance of C and D; when if A and C were applied to one another, B and D might also be applied to one another at the same time.
XII. Two or more points, as B, C, D, are said to be equidistant
from another point A, when B and A, C and A, D and A, are equally distant each pair with the other.
XIII. Figures in general, as being things capable of being com
pared in point of greatness with objects of their own kind, [that is to say, solids with solids, surfaces with surfaces, fc.], are called magnitudes.
XIV. Magnitudes which if their boundaries were applied to one
another, would coincide; or might be made capable of doing
1. Those whose boundaries can actually be applied together and
be in contact in every part at once; as a cast to its mould, a stamp to its impression, or one side of an indenture to the other.
Which are called reciprocals. 2. Those whose boundaries it can be shown would coincide, if
they could be applied to one another without bar of corporeal substance,—though they cannot be actually applied, by reason
of such bar. Which are called fac-similes. 3. Such as might be reduced to reciprocals or fac-similes, by a
different arrangement of parts. Which are called equivalents.
XV. A magnitude is said to be greater than another by a
certain magnitude, when this last-mentioned magnitude being taken from the first, the remainder is equal to the second. And a magnitude is said to be less than another by a certain magnitude, when this last-mentioned magnitude being added to the
first, the sum is equal to the second. XVI. Magnitudes are said to be given, when equals to them
can be assigned. XVII. The science which treats of the relations and properties
of magnitudes, is named Geometry. XVIII. An assertion which it is proposed to show to be true,
is called a Theorem. An operation which it is proposed to show how to perform, is called a Problem. And both are called
by the title of Propositions. XIX. A Proposition dependent on some other that has been
previously established; but which, by reason of the simplicity of the steps required to connect it with the former, or other causes, it is convenient to state without the formalities of a
separate Proposition; is called a Corollary. XX. A Proposition of which the establishment is necessary to
the establishment of some other ; but which, by reason of its being taken from a different branch of science, or other causes, it is convenient to distinguish by a separate title, rather than to number among the Propositions in the ordinary way; is called a Lemma.
XXI. A remark or observation connected with something that has
gone before, is called a Scholium.
XXII. When a thing is said to be so and so by the hypothesis,
the meaning is, that its being so and so, makes part of the original supposition or statement of the question in hand. Thus, if the proposition is, that if two magnitudes be equal, some
particular result shall follow, as, for instance, that the sum of them shall be equal to some third magnitude; one of these magnitudes may at any period of the operations in hand be avoyched to be equal to the other by the hypothesis. For that they are equal, is the original supposition or groundwork on which the whole question proceeds.
XXIII. When a thing is said to be so and so by construction,
the meaning is, that it has at some previous period of the operations in hand, been made to be what it is said to be. Thus, if in the course of the operations in hand, some magnitude has
been cut off equal to another magnitude; these two magnitudes may at any time afterwards be avouched to be equal by construction. For one of them has been specially constituted and constructed
equal to the other. XXIV. When a thing is said to be so and so by parity of
reasoning, the meaning is, that what has been shown to be true in some previous instance, may by taking the same steps be shown be in this. Thus, if it has been shown that because two particular magnitudes
are each equal to a third magnitude they are equal to one another ; and if the same steps can be applied in any other instance ; it may be avouched that by parity of reasoning any other two magnitudes which are equal to a third, are equal to one another. For it is open
to be demonstrated by the same steps. When the parity of reasoning is extended to all instances to which the
proposition is capable of being applied, (or in other words, to all which come under its terms], the proposition is said to be established universally. And a proposition so established is called a universal
proposition. XXV. A conclusion the truth of which is shown to be so con
nected with the truth of some preceding position or statement, that the preceding cannot be true, without the other being true
also ; is called a consequence. XXVI. When such preceding position is not known to be true,
but is only assumed to be true for the purpose of trying its truth or falsehood by examination of the consequences, it is called an assumption. And because the assumption cannot be true without the consequence being true also; if the assumption is found to involve a false or impossible consequence, the assumption cannot be true. This is the rationale (or reasonable principle] of the mode of proof im
properly called reductio ad absurdum, "reduction to an absurdity.' XXVII. The establishment of some universal proposition, is called a demonstration. The peculiar object of a demonstration, is to display the reasons which
make a certain proposition necessarily true in all the instances that can by possibility come under its terms; in contradistinction to such