Euclid's Elements of geometry, the first six books, chiefly from the text of Dr. Simson, with explanatory notes; a series of questions on each book ... Designed for the use of the junior classes in public and private schools online

. (page 24 of 38)
Online LibraryEuclidEuclid's Elements of geometry, the first six books, chiefly from the text of Dr. Simson, with explanatory notes; a series of questions on each book ... Designed for the use of the junior classes in public and private schools → online text (page 24 of 38)
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greatest magnitude which will measure both of them, it is called the
greatest common measure of the two magnitudes : also when two magni-
tudes of the same kind have no common measure, they are said to be
incommensurable. The same terms are also applied to numbers.

Unity has no magnitude, properly so called, but may represent that
portion of every kind of magnitude which is assumed as the measure of
all magnitudes of the same kind , The composition of unities cannot pro-
duce Geometrical magnitude ; three units are more in number than one
unit, but still as much different from magnitude as unity itself. Numbers
may be considered as quantities, for we consider every thing that can be
exactly measured, as a quantity.

Unity is a common measure of all rational numbers, and all numerical
reasonings proceed upon the hypothesis that the unit is the same through-
out the whole of any particular process. Euclid has not fixed the magni-
tude of any unit of length, nor made reference to any unit of measure of
lines, surfaces, or volumes. Hence arises an essential diff'erence between
number and magnitude ; unity, being invariable, measures all rational
numbers ; but though any quantity be assumed as the unit of magnitude,
it is impossible to assert that this assumed unit will measure all other
magnitudes of the same kind.


All whole numbers therefore are commensurable ; for unity is their
common measure r also all rational fractions proper or improper, are com-
mensurable ; for any such fractions may be reduced to other equivalent
fractions having one common denominator, and that fraction whose de-
nominator is the common denominator, and whose numerator is unity,
will measure any one of the fractions. Two magnitudes having a common
measure can be represented by two numbers which express the number of
times the common measure is contained in both the magnitudes.

But two incommensurable magnitudes cannot be exactly represented by
any two whole numbers or fractions whatever ; as, for instance, the side
of a square is incommensurable to the diagonal of the square. For, it may
be shewn numerically, that if the side of the square contain one unit of
length, the diagonal contains more than one, but less than two units of
length. If the side be divided into 10 units, the diagonal contains more
than 14, but less than 15 such units. Also if the side contain 100 units,
the diagonal contains more than 141, but less than 142 such units. It is
also obvious, that as the side is successively divided into a greater number
of equal parts, the error in the magnitude of the diagonal will be diminished
continually, but never can be entirely exhausted ; and therefore into what-
ever number of equal parts the side of a square be divided, the diagonal
will never contain an exact number of such parts. Thus the diagonal and
side of a square having no common measure, cannot be exactly repre-
sented by any two numbers.

The terra equimultiple in Geometry is to be understood of magnitudes
of the same kind, or of different kinds, taken an equal number of times, and
implies only a division of the magnitudes into the same number of equal
parts. Thus, if two given lines are trebled, the trebles of the lines are
equimultiples of the two lines : and if a given line and a given triangle be
trebled, the trebles of the^ine and triangle are equimultiples of the line
and triangle: as (vi. 1. fig.) the straight line HC and the triangle AHC
are equimultiples of the line BC and the triangle ABC: and in the same
manner, (vi. 33. fig.) the arc EN and the angle EHN are equimultiples of
the arc EF and the angle EHF.

Def. III. Ao'yos- icTTL duo fxtytduiv ofioysvcvu i] Kara 'TrrjXi/cdxjjTa tt/oo?
d\Xi]Xa iroid o-x£<Tts. By this definition of ratio is to be understood the con-
ception of the mutual relation of two magnitudes of the same kind, as two
straight lines, two angles, two surfaces, or two solids. To prevent any
misconception, Def. iv. lays down the criterion, whereby it may be known
what kinds of magnitudes can have a ratio to one another ; namely,
Aoyov BX^'-i' -Jrpo'i d\.\r\\a /jLtyidi] Xg'ysTat, a duuuTai Tro/WaTrA.acria^d/ifj/a
d\Xi]Xu)v vTTEpsx^i-i^' " Magnitudes are said to have a ratio to one another,
which, when they are multiplied, can exceed one another ;" in other w^ords,
the magnitudes which are capable of mutual comparison must be of the
same kind. The former of the two terms is called the antecedent ; and the
latter, the consequent of the ratio. If the antecedent and consequent are
equal, the ratio is called a ratio of equality ; but if the antecedent be greater
or less than the consequent, the ratio is called a ratio of greater or of less
inequality. Care must be taken not to confound the ex])rfcssions " ratio
of equality", and •' equality of ratio :" the former is applied to the terms
of a ratio when they, the antecedent and consequent, are equal to one
another, but the latter, to two or more ratios, when they are equal.

Arithmetical ratio has been defined to be the relation which one number
bears to another with respect to quotity ; the comparison being made by
considering what multiple, part or parts, one number is of the other.


An arithmetical ratio, therefore, is represented by the quotient which
arises from dividing the antecedent by the consequent of the ratio ; or by
the fraction which has the antecedent for its numerator and the consequent
for its denominator. Hence it will at once be obvious that the properties
of arithmetical ratios will be made to depend on the properties of fractions.

It must ever be borne in mind that the subject of Geometry is not
number, but the magnitude of lines, angles, surfaces, and solids ; and its
object is to demonstrate their properties by a comparison of their absolute
and relative magnitudes.

Also, in Geometry, multiplication is only a repeated addition of the same
magnitude ; and division is only a repeated subtraction, or the taking of a
less magnitude successively from a greater, until there be either no re-
mainder, or a remainder le|| than the magnitude which is successively

The Geometrical ratio of any two given magnitudes of the same kind
will obviously be represented by the magnitudes themselves ; thus, the
ratio of two lines is represented by the lengths of the lines themselves;
and, in the same manner, the ratio of two angles, two surfaces, or two
solids, will be properly represented by the magnitudes themselves.

In the definition of ratio as given by Euclid, all reference to a third
magnitude of the same geometrical species, by means of which, to compare
the two, whose ratio is the subject of conception, has been carefully
avoided. The ratio of the two magnitudes is their relation one to the other,
without the intervention of any standard unit whatever, and all the pro-
positions demonstrated in the Fifth Book respecting the equality or i7ie-
quality of two or more ratios, are demonstrated independently of any know-
ledge of the exact numerical measures of the ratios ; and their generality
includes all ratios, whatever distinctions may be made, as to the terms of
them being commensurable or incommensurable.

In measuring any magnitude, it is obvious that a magnitude of the
same kind must be used ; but the ratio of two magnitudes may be measured
by every thing which has the property of quantity. Two straight lines
will measure the ratio of two triangles, or parallelograms (vi. 1. fig.) : and
two triangles, or two parallelograms will measure the ratio of two straight
lines. It would manifestly be absurd to speak of the line as measuring
the triangle, or the triangle measuring the line. (See notes on Book it.)

The ratio of any two quantities depends on their relative and not their
absolute magnitudes ; and it is possible for the absolute magnitude of two
quantities to be changed, and their relative magnitude to continue the
same as before ; and thus, the same ratio may subsist between two given
magnitudes, and any other two of the same kind.

In this method of measuring Geometrical ratios, the measures of the
ratios are the same in number as the magnitudes themselves. It has how-
ever two advantages ; first, it enables us to pass from one kind of magni-
tude to another, and thus, independently of any numerical measure, to
institute a comparison between such magnitudes as cannot be directly
compared with one another : and secondly, the ratio of two magnitudes
of the same kind may be measured by two straight lines, which form a
simpler measure of ratios than any other kind of magnitude.

But the simplest method of all would be, to express the measure of the
ratio of tico magnitudes by one ; but this cannot be done, unless the two
magnitudes are commensurable. If two lines AB, CD, one of which AB
contains 12 units of any length, and the other CD contains 4 units of the
same length ; then the ratio of the line AB to the line CD^ is the same as the

238 Euclid's elements.

ratio of the number 12 to 4. Thus, two numbers may represent the ratio
of two lines when the lines are commensurable. In the same manner, two
numbers may represent the ratio of two angles, two surfaces, or two solids.

Thus, the ratio of any two magnitudes of the same kind may be ex-
pressed by two numbers, when the magnitudes are commensurable. By
this means, the consideration of the ratio of two magnitudes is changed to
the consideration of the ratio of two numbers, and when one number is
divided by the other, the quotient will be a single number, or afractioriy
which will be a measrire of the ratio of the two numbers, and therefore of
the two quantities. If 12 be divided by 4, the quotient is 3, which mea-
Bures the ratio of the two numbers 12 and 4. Again, if besides the ratio
of the lines AB and CD which contain 12 and 4 units respectively, we con-
sider two other lines £Fand G// which contain 9 and 3 units respectively ;
it is obvious that the ratio of the line EF to GH is the same as the ratio
of the number 9 to the number 3. And the measure of the ratio of 9 to
3 is 3. That is, the numbers 9 and 3 have the same ratio as the numbers
12 and 4.

But this is a numerical measure of ratio, and can only be applied strictly
when the antecedent and consequent are to one another as one number to

And generally, if the two lines AB, CD contain a and b units respec-
tively, and q be the quotient which indicates the number of times the
number b is contained in a, then q is the measure of the ratio of the two
numbers a and b : and if EF and GH contain c and d units, and the number
d be contained q times in c : the number a has to b the same ratio as the
number c has to d.

This is the numerical definition of proportion, which is thus expressed
in Euclid's Elements, Book vii, definition 20. " Four numbers are pro-
portionals when the first is the same multiple of the second, or the same
part or parts of it, as the third is of the fourth." This definition of the
proportion of four numbers, leads at once to an equation :

for, since a contains 6, q times \ - = q;



and since c contains d, q times ; - = <7 :


therefore •; = -, which is the fundamental equation upon which all the

b d
reasonings on the proportion of numbers depend.

If four numbers be proportionals, the product of the extremes is equal
to the product of the means.

For if a, b, c, d be proportionals, or a : b i: c i d.

Then 3 = 5;

Multiply these equals by bd,

ahd chd
•*• ~b' ^~d*
or, ad = bCy
that is, the product of the extremes is equal to the product of the means.
And conversely, If the product of the two extremes be equal to the
product of the two means, the four numbers are proportionals.
For if a, b, c, d, be four quantities,


such that ad = bcy

a c

then dividing these equals by bd^ therefore - =

h d

and a '. h :'. c '. d^
or the first number has the same ratio to the second, as the third has to
the fourth.

If c = 6, then ad = b^; and conversely if ad = b^ : then - = -, .


These results are analogous to Props. 16 and 17 of the Sixth Book.

Sometimes a proportion is defined to be the equality of two ratios.

Def. VIII declares the meaning of the term analogy or proportion. The
ratio of two lines, two angles, two surfaces or two solids, means nothing
more than their relative magnitude in contradistinction to their absolute
magnitudes ; and a similitude or likeness of ratios implies, at least, the two
ratios of the four magnitudes which constitute the analogy or proportion.

Def. IX states that a proportion consists in three terms at least; the
meaning of which is, that the second magnitude is repeated, being made
the consequent of the first, and the antecedent of the second ratio. It is
also obvious that when a proportion consists of three magnitudes, all three
are of the same kind. Def. vi appears only to be a further explanation
of what is implied in Def. viii.

Def. v. Proportion having been defined to be the similitude of ratios^
or more properly, the equality or identity of ratios, the fifth definition lays
down a criterion by which two ratios may be known to be equal, or four
magnitudes proportionals, without involving any inquiry respecting the
four quantities, whether the antecedents of the ratios contain or are con-
tained in their consequents exactly ; or whether there are any magnitudes
which measure the terms of the two ratios. The criterion only requires,
that the relation of the equimultiples expressed should hold good, not
merely for any particular multiples, as the doubles or trebles, but for any
multiples whatever, whether large or small.

This criterion of proportion may be applied to all Geometrical magni-
tudes which can be multiplied, that is, to all which can be doubled, trebled,
quadrupled, &c. But it must be borne in mind, that this criterion does
not exhibit a definite measure for either of the two ratios which constitute
the proportion, but only, an undetermined measure for the sameness or
equality of the two ratios. The nature of the proportion of Geometrical
magnitudes neither requires nor admits of a numerical measure of either
of the two ratios, for this would be to suppose that all magnitudes are
commensurable. Though we know not the definite measure of either of
the ratios, further than that they are both equal, and one may be taken as
the measure of the other, yet particular conclusions may be arrived at by
this method : for by the test of proportionality here laid down, it can be
proved that one magnitude is greater than, equal to, or less than another :
that a third proportional can be found to two, and a fourth proportional
to three straight lines, also that a mean proportional can be found be-
tween two straight lines : and further, that which is here stated of
straight lines may be extended to other Geometrical magnitudes.

The fifth definition is that of equal ratios. The definition of ratio itself
(defs. 3, 4) contains no criterion by which one ratio may be known to be
equal to another ratio ; analogous to that by which one magnitude is
known to be equal to another magnitude (Euc. i. Ax. 8). The preceding
definitions (3, 4) only restrict the conception of ratio within certain limits,

240 Euclid's elements.

but lay down no test for comparison, or tlie deduction of properties. All
Euclid's reasonings were to turn upon this comparison of ratios, and
hence it was competent to lay down a criterion of equality and inequality
of two ratios between two pairs of magnitudes. In short, his effective de-
finition is a definition of proportionals.

The precision with which this definition is expressed, considering the
number of conditions involved in it, is remarkable. Like all complete
definitions the terms (the subject and predicate) are convertible : that is,

(a) If four magnitudes be proportionals, and any equimultiples be
taken as prescribed, they shall have the specified relations with respect
to ** greater, greater," &c.

{b) If of four magnitudes, two and two of the same Geometrical
Species, it can be shewn that the prescribed equimultiples being taken,
the conditions under which those magnitudes exist, must he such as to
fulfil the criterion *' greater, greater, &c." ; then these four magnitudes
shall be proportionals.

It may be remarked, that the cases in which the second part of the
criterion (" equal, equal"; can be fulfilled, are comparatively few: namely
those in which the given magnitudes, whose ratio is under consideration,
are both exact multiples of some third magnitude — or those which are
called commensurable. When this, however, is fulfilled, the other two will
be fulfilled as a consequence of this. When this is not the case, or the
magnitudes are incommensurable ^ the other two criteria determine the pro-
portionality. However, when no hypothesis respecting commensur-
ability is involved, the contemporaneous existence of the three cases
(•' greater, greater; equal, equal ; less, less") must be deduced from the
hypothetical conditions under which the magnitudes exist, to render the
criterion valid.

With re$pect to this test or criterion of the proportionality of four
magnitudes, it has been objected, that it is utterly impossible to make
trial of all the possible equimultiples of the first and third magnitudes,
and also of the second and fourth. It may be replied, that the point in
question is not determined by making such trials, but by shewing from
the nature of the magnitudes, that whatever be the multipliers, if the
multiple of the first exceeds the multiple of the second magnitude, the
multiple of the third will exceed the multiple of the fourth magnitude,
and if equal, icill be equal ; and if less, will be less, in any case which
may be taken.

The Ai-ithmetical definition of proportion in Book vii, Def. 20, even
if it were equally general with the Geometrical definition in Book v, Def.
6, is by no means universally applicable to the subject of Geometrical
magnitudes. The Geometrical criterion is founded on multiplication,
which is always possible. When the magnitudes are commensurable, the
multiples of the first and second may be equal or unequal ; but when th^
magnitudes are incommensurable, any multiples whatever of the first and
second mwsi be unequal ; but the Arithmetical criterion of proportion is
founded on division, which is not always possible. Euclid has not shewn
in Book v, how to take any part of a line or other magnitude, or that the
tAvo terms of a ratio have a common measure, and therefore the numerical
definition could not be strictly applied, even in the limited way in which
it may be applied.

Number and Magnitude do not correspond in all their relations ; and
hence the distinction between Geometrical ratio and Arithmetical ratio ;
the former is a comparision /card TrjjXtKOTTjra, according to quantity, but


the latter, according to quotity. The former gives an undetermined,
though definite measure, in magnitudes ; but the latter attempts to
give the exact value in numbers.

The fifth book exhibits no method whereby two magnitudes may be
determined to be commensurable, and the Geometrical conclusions de-
duced from the multiples of magnitudes are too general to furnish a
numerical measure of ratios, being all independent of the commensura-
bility or incommensurability of the magnitudes themselves.

It is the numerical ratio of two magnitudes which will more certainly
discover whether they are commensurable or incommensurable, and
hence, recourse must be had to the forms and properties of numbers.
All numbers and fractions are either rational or irrational. It has been
seen that rational numbers and fractions ca7i express the ratios of Geo-
metrical magnitudes, when they are commensurable. Similar relations
( f incommensurable magnitudes may be expressed by irrational numbers,
li' the Algebraical expressions for such numbers may be assumed and
emploved in the same manner as rational numbers. The irrational
expressions being considered the exact and definite, though undeter-
mined, values of the ratios, to which a series of rational numbers may
successively approximate.

Though two incommensurable magnitudes have not an assignable nume«
rical ratio to one another, yet they have a certain definite ratio to one
another, and two other magnitudes may have the same ratio as the first
two : and it will be found, that, when reference is made to the numerical
value of the ratios of four incommensurable magnitudes, the same irra-
tional number appears in the two ratios.

The sides and diagonals of squares can be shewn to be proportionals,
and though the ratio of the side to the diagonal is represented Geome-
trically by the two lines which form the side and the diagonal, there is
no rational number or fraction which will measure exactly their ratio.

If the side of a square contain a units, the ratio of the diagonal to the
side is numerically as V 2 to 1 ; and if the side of another square contain
b units, the ratio of the diagonal to the side will be found to be in the
ratio of V 2 10 1. Again, the two parts of any number of lines which
may be divided in extreme and mean ratio will be found to be respectively
m the ratio of the irrational number V5 — 1 to 3 — VS. Also, the
ratios of the diagonals of cubes to the diagonals of one of the faces will
be found to be in the irrational or incommensurate ratio of V 3 to v/ 2.

Thus it will be found that the ratios of all incommensurable magni-
tudes which are proportionals do involve the same irrational numbers,
and these may be used as the numerical measures of ratios in the same
manner as rational numbers and fractions.

It is not however to such enquiries, nor to the ratios of magnitudes
when expressed as rational or irrational numbers, that Euclid's doctrine
of proportion is legitimately directed. There is no enquiry into what a
ratio is in numbers, but whether in diagrams formed according to assigned
conditions, the ratios between certain parts of the one are the same as
the ratios between corresponding parts of the other. Thus, with respect
to any two squares, the question that properly belongs to pure Geometry
is : — whether the diagonals of two squares have the same ratio as the
sides of the squares? Or whether the side of one square has to its
diagonal, the same ratio as the side of the other square has to its diagonal?
Or again, whether in Euc. vi. 2, Avhen BC and DE are parallel, the line
BD has to the line D^, the same ratio that the line CE has to the line



AE ? There is no purpose on the part of Euclid, to assign either of these
ratios in tmmbers: but only to prove that their universal sameness is
inevitably a consequence of the original conditions according to which
the diagrams were constituted. There is, consequently, no introduction
of the idea of incommensurables : and indeed, with such an object as
Euclid had in view, the simple mention of them would have been at least
irrelevant and superfluous. If however it be attempted to apply numeri-
cal considerations to pure geometrical investigations, incommensurables
will soon be apparent, and difficulties will arise which were not foreseen.
Euclid, however, effects his demonstrations without creating this arti-
ficial difficulty, or even recognising its existence. Had he assumed a
standard unit of length, he would have involved the subject in numeri-
cal considerations ; and entailed upon the subject of Geometry the
almost insuperable difficulties which attach to all such methods.

It cannot, however, be too strongly or too frequently impressed upon
the learner's mind, that all Euclid's reasonings are independent of the
numerical expositions of the magnitudes concerned. That the enquiry
as to what numerical function any magnitude is of another, belongs not
to Pure Geometry, but to another Science. The consideration of any
intermediate standard unit does not enter into p\ire Geometry ; into
Algebraic Geometry it essentially enters, and indeed constitutes the funda-
mental idea. The former is wholly free from numerical considerations ;
the latter is entirely dependent upon them.

Def. VII is analogous to Def. 5, and lays down the criterion whereby
the ratio of two magnitudes of the same kind may be known to be greater
or less than the ratio of two other magnitudes of the same kind.

Def. XI includes Def. x. as three magnitudes may be continued pro-
portionals, as well as four or more than four. In continued proportionals,
all the terms except the first and last, are made successively the conse-

Online LibraryEuclidEuclid's Elements of geometry, the first six books, chiefly from the text of Dr. Simson, with explanatory notes; a series of questions on each book ... Designed for the use of the junior classes in public and private schools → online text (page 24 of 38)