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or if

and in Del. 7 what we have to understand by a greater or a less ratio. I then a te is the same multiple of B as c+f is of d. viz..
The 6th definition is only nominal, explaining the meaning of the
word proportional.

eter(m+n)b, and c+f=(m+n)d. Euclid represents magnitudes by lines, and often denotes them Prop. 3. !! a=mb, c-md, then is na the same multiple of b either by single letters or, like lines, by two letters. We shall use that nc is of d, viz. na =nmb, nc = nmd. only single letters for the purpose. If a and b denote two magnitudes of the same kind; their ratio will be denoted by a : b; if c and d are

Prop. 4. If

Q:6::c:d, then

ma : nb :: mc : nd. two other magnitudes of the same kind, but possibly of a different kind from a and b, then if c and d have the same ratio to one another Prop. 5. If

a=mb, and c=md, as a and 6, this will be expressed by writing


a-c=m(b-d). Q:6::c:d.

Prop. 6. Ir

a=mb, c=md, Further, if m is a (whole), number, mo shall denote the multiple then are a-nb and c-nd either equal to, or equimultiples of, o of a which is obtained by taking it m times.

and d, viz. a - nb = (m-n)b and cond=(m-n)d, where m-n may 49. The whole theory of ratios is based on Def. 5.

be unity. Def. 5. The first of four magnitudes is said to have the same ratio

All these propositions relate to equinultiples. . Now follow proto the second that the third has to the fourth when, any equimultiples positions about ratios which are compared as to their magnitude. whalever of the first and the third being taken, and any, equimultiplos $ 52. Prop: 7. If a =b, then a :c7:6: cand c:a ::c: b. whalever of the second and the fourth, the multiple of the first be less The prool is simply this. As a = b we know that ma = mb; there. than that of the second, the multiple of the third is also less than that of fore il

ma>nc, then mb>nc, the fourth; and if the multiple of the first is equal to that of the second, if

ma=nc, then mb=nc, the multiple of the third is also equal to that of the fourth; and if the if multiple of the first is greater than that of the second, the multiple of therefore the first proportion holds by Definition 5.

ma <nc, then mb<nc, the third is also greater than that of the fourth. It will be well to show at once in an example how this definition

Prop. 8. If a>b, then a :c>6.:9, can be used, by proving the first part of the first proposition in the and

c:esc:6. sixth book. Triangles of the same altitude are lo one another as their bases, or if a and b are the bases, and a and 8 the areas, of two

The proof depends on Definition 7, triangles which have the same altitude, then a : 6::a: 8.

Prop. 9 (converse to Prop. 7). If To prove this, we have, according to Definition 5, to show

Q:c::b:c, if ma >nb, then ma >nB,

C:2::c:b, then a=b. if ma=nb, then ma=nb.

Prop. 10 (converse to Prop. 8). If if ma <nb, then ma <nB.

a:c>b:c, then a> That this is true is in our case easily seen. We may suppose that and if

6:2<c:b, then a<b. the triangles have a common vertex, and their bases in the same line. We set off the base e along the line containing the bases Prop. 11. II


and m times; we then join the different parts of division to the vertex,

0:6::e:f. and get m triangles all equal to a. The triangle on ma as base equals,


c:d::e:f. therefore, ma. If we proceed in the same manner with the base b, In words, if two ralios are equal to a third, they are equal to one setting it off n times, we find that the area of the triangle on the another. After these propositions have been proved, we have a base nb equals nb, the vertex of all triangles being the same. But right to consider a ratio as a magnitude, for only pow can we con if two triangles have the same altitude, then their areas are equal sider a ratio as something for which the axiom about magnitudes if the bases are equal; hence ma=nB il ma = nb, and if their bases holds: things which are equal to a third are equal to one another. are unequal, then that has the greater area which is on the greater We shall indicate this by writing in future the sign = instead base; in other words, ma is greater than, .equal to, or less than of ::. The remaining propositions, which explain themselves, may nB, according as ma is greater than, equal to, or less than nb, which then be stated as follows: was to be proved. 50. It will be seen that even in this example it does not become

$ 53. Prop. 12. If @: b=c:dee:f. evident what a ratio really is. It is still an open guestion whether

a+cte: 6+d+f=Q : b. ratios are magnitudes which we can compare. We do not know

Prop. 13. I a: b=c:d and c: d>e : f, whether the ratio of two lines is a magnitude of the same kind as the then

a:b>e : f. ratio of two areas. Though we might say that Def. 5 defines equal ratios, still we do not know whether they are equal in the sense of Prop. 14. If a:bc:d, and a>c, then b>d. the axiom, that two things which are equal to a third are equal to Prop. 15. Magnitudes have the same ratio to one another that one another. That this is the case requires a proof, and until this their equimultiples haveproof is given we shall use the :: instead of the sign =, which, how

ma : mb=Q : b. ever, we shall afterwards introduce. As soon as it has been established that all ratios are like magni

Prop. 16. If a, b, c, d are magnitudes of the same kind, and if tudes, it becomes easy to show that, in some cases at least, they

Q: b=c:d,

then are numbers. This step was never made by Greek mathematicians.

Q:c=b:d. They distinguished always most carefully between continuous Prop. 17. II

a+: b=c+did, magnitudes and the discrete series of numbers. In modern times then

a: b=c:d. it has become the custom to ignore this difference.

Prop. 18 (converse to 17). If If, in determining the ratio of two lines, a common measure can

Q:bc:d be found, which is contained m times in the first, and n times in

then the second, then the ratio of the two lines equals the ratio of the

a+b: b=c+d:d. two numbers m : n. This is shown by Euclid in Prop. 5, X. But the

Prop. 19. If a, b, c, d are quantities of the same kind, and if ratio of two numbers is, as a rule, a fraction, and the Greeks did

a:b=cid, not, as we do, consider fractions as numbers. Far less had they then

a-c:b-d=Q :b. any notion of introducing irrational numbers, which are neither whole nor fractional, as we are obliged to do if we wish to say that $ $4: Prop. 20. If there be three magnitudes, and another three, all ratios are numbers. The incommensurable numbers which are

which have the same ralio, laken two and two, then if the first be greater thus introduced as ratios of incommensurable quantities are nowa

than the third, the fourth shall be greater than the sixth; and if equal, days as familiar to us as fractions; but a proof is generally omitted equal; ond if less, less. that we may apply to them the rules which have been established If we understand by for rational numbers only. Euclid's treatment of ratios avoids this

a:0:0:0:e:... =':':c": d:d:.. difficulty. His definitions hold for commensurable as well as for that the ratio of any two consecutive magnitudes on the first side incommensurable quantities. Even the notion of incommensurable equals that of the corresponding magnitudes on the second side, quantities is avoided in Book V. But he proves that the more we may write this theorem in symbols, thus: elementary rules of algebra hold for ratios. We shall state all If a, b, c be quantities of one, and d, e, f magnitudes of the same his propositions in that algebraical form to which we are now accus. or any other kind, such that tomed. This may, of course, be done without changing the character of Euclid's method.

a:0:c=d:6:3 $ 51. Using the notation explained above we express the first and if

a>c, then d>f, propositions as follows:

but if

a=c, then daj,

and if Prop. 1. If Q = ma', b=m8, c=mc',

a<c, then d<f. then a+b+c=m(a' 16'+c).

Prop. 21.11 a:b=e : f and b:c=d:e, Prop. 2. If a - mb, and c=ind,

or if e=nb, and frnd,




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and if
a>, then d>f.

2. (Prop. s). If the sides of the one are proportional to those of but if 0 =C, then dos,

the other; and if acc, then d.

3. (Prop. 6). If two sides in one are proportional to two sides in By aid of these two propositions the following two are proved.

the other, and if the angles contained by these sides are equal;

4. (Prop. 7). If two sides in one are proportional to two sides in $ 55. Prop. 22: 1) there be any number of magnitudes, and as many others, which have the same ratio, taken two and two in order,

the other, if the angles opposite homologous sides are equal, and if the first shall have to the last of the first magnitudes the same ralio

the angles opposite the other homologous sides are both acute, both right

or both obluse; homologous sides being in each case those which are which the first of the others has to the last.

opposite equal angles. We may state it more generally, thus:

An important application of these theorems is at once made to If Q:b:cid:e:... =a' : b':d : d':e:...,

a right-angled triangle, viz. = then not only have two consecutive, but any two magnitudes on Prop. 8. In a righe-angled triangle, if a perpendicular be drawn the first side, the same ratio as the corresponding magnitudes on from the right angle to the base, the triangles on each side of il are the other. For instance

similar to the whole triangle, and to one another. a:c=c':0;6:e=b' :e', &c.

Corollary: -From this it is manisest that the perpendicular Prop. 23 we state only in symbols, viz:

drawn from the right angle of a right-angled triangle to the base

is a mean proportional between the segments of the base, and also If. 0:0:0:0:e:... 1:51...,

that each of the sides is a mean proportional between the base and

the segment of the base adjacent to that side. then a:c=c' :a',

$61. There follow four propositions containing problems, in b:e=e' : 6',

language slightly different from Euclid's, viz. :and so on.

Prop. 9.

To divide a straight line into a given number of equal Prop. 24 comes to this : Ifa : b=c:d and e :b=f:d, then


Prop. 10. To divide a straight line in a given ratio. ate:b=c+f:d.

Prop. 11. To find a third proportional to two given straight lines. Some of the proportions which are considered in the above pro

Prop. 12. To find a fourth proportional to three given straight positions have special names. These we have omitted, as being of lines. no use, since algebra has enabled us to bring the different operations Prop. 13. To find a mean proportional between two given straigkt contained in the propositions under a common point of view.


The last three may be written as equations with one unknown character.

quantity-viz. if we call the given straight lines a, b, c, and the Prop. 25. If four magnitudes of the same kind be proportional

, required line <, we have to find a line x so that the grealesi and least of them together shall be greater than the other two together. In symbols-.

Prop. II.

a:b=:x; If a, b, c, d be magnitudes of the same kind, and if a : b=c:d, Prop. 12.

a:bc:*; and if a is the greatest, henced the least, then atd>6+c.

Prop. 13

0:*=* :b. 57. We return once again to the question, What is a ratio ? We shall see presently how these may be written without the We have seen that we may treat ratios as magnitudes, and that all signs of ratios. ratios are magnitudes of the same kind, for we may compare any $62. Euclid considers next proportions connected with parallelatwo as to their magnitude. It will presently be shown that ratios grams and triangles which are equal in area. of lines may be considered as quotients of lines, so that a ratio appears Prop. 14. Equal parallelograms which have one angle of the one as answer to the question, How often is one line contained in another? equal to one angle of the other have their sides about the equal angles But the answer to this question is given by a number, at least in reciprocally proportional;, and parallelograms which have one angle some cases, and in all cases if we admit incommensurable numbers. of the one equal to one angle of the olher, and their sides about the equal Considered from this point of view, we may say the fifth book of the angles reciprocally proportional, are equal to one another. Elements shows that some of the simpler algebraical operations Prop. 15. Equal triangles which have one angle of the one equal hold for incommensurable numbers. In the ordinary algebraical to one angle of the other, have their sides about the equal angles reciprotreatment of numbers this proof is altogether omitted, or given by cally proportional; and triangles which have one angle of the one equal a process of limits which does not seem to be natural to the subject. to one angle of the other, and iheir sides about the equal angles recipro.

cally proportional, are equal to one another. BOOK VI.

The latter proposition is really the same as the former, for is, as $58. The sixth book contains the theory of similar figures in the figure belonging to the

in the accompanying diagram, After a few definitions explaining terms, the first proposition gives former the two equal parallelo

A the first application of the theory of proportion.

grams AB and BC be bisected Prop. 1. Triangles and parallelograms of the same altitude are to

by the lines DF and EG, and one another as their bases.

if EF be drawn, we get the

The proof has already been considered in $ 49.
From this follows easily the important theorem

figure belonging to the latter.

It is worth noticing that Prop. 2. If a straight line be drawn parallel to one of the sides the lines fe and DG are of a triangle'il shall cut the other sides, or those sides produced. P10- parallel. We may state thereportionally; and if the sides or the sides produced be cut proportionally, lore the theorem the straight line which joins the points of section shall be parallel lo

If two triangles are equal in the remaining side of the triangle.

area, and have one angle in the one vertically opposite to one angle $_59. The next proposition, together with one added by Simson

in the other, then the two straight lines which join the remaining two as Prop. A, may be expressed more conveniently if we introduce a modern phraseology, viz. is in a line AB we assume a point C between

vertices of the one to those of the other triangle are parallel. A and B, we shall say that C divides AB internally in the ratio

$63. Å most important theorem is AC : CB; but if C be taken in the line AB produced, we shall say

Prop. 16. If four straight lines be proportionals, the retangle that AB is divided externally in the ratio AC :CB.

contained by the extremes is equal to the rectangle contained by the The two propositions then come to this:

means;, and if the rectangle contained by the extremes be equal to the Prop. 3. The bisector of an angle in a triangle divides the opposite rectangle

contained by the means, the four straight

lines are proportionals. side internally in a ratio equal to the ratio of the two sides including

In symbols, il a, b, c, d are the four lines, and if

a:b=c:d, That angle; and conversely, if a line through the vertex of a triangle


ad=bc; divide ihe base internally in the ratio of the two other sides, then that

and conversely, if

ad=bc, line bisects the angle at the verlex.

then Simson's Prop. A. The line which bisects an exterior angle of a


where ad and bc denote (as in $ 20), the areas of the rectangles Iriangle divides the opposite side externally in the ratio of the oiker sides; and conversely, if a line through the verlex of a triangle divide

contained by a and d and by b and c respectively: the base externally in the ratio of the sides, then it bisects en exterior into an equation between two products.

This allows us to transform every proportion between four lines angle at the verlex of the triangle. If we combine both we have

It shows further that the operation of forming a product of two The two lines which bisect the interior and exterior angles at one lines, and the operation of forming their ratio are each the inverse vertex of a triangle divide the opposite side internally and externally of the other. in the same ratio, viz.

in the ralio of the other two sides. $ 60. The next four propositions contain the theory of similar

If we now define a quotient of two lines as the number which triangles, of which four cases are considered. They may be stated multiplied into b gives a, so that together.

Two triangles are similar -
1. (Prop. 4). If the triangles are equiangular:


we see that from the equality of two quotients

call this number a the numerical value of a. If in the saine manner

B be the numerical value of a line o we have %-á

Q:ba:B; follows, if we multiply both sides by bd,

in words: The ratio of two lines (and of two like quantities in general)

is equal to that of their numerical values. go.d = d.o.

This is easily proved by observing that a =au, b = Bu, therefore

a: b=au: Bu, and this may without difficulty be shown to equal a :B. ad cb.

If now a, 6 be base and altitude of one, a', b' those of another But from this it follows, according to the last theorem, that parallelogram, , B and a', B' their numerical values respectively,

and A, A' their areas, then a:b=c:d

A Qbasaß. Hence we conclude that the quotients and the ratio a : 6 are

A P788 different forms of the same magnitude, only with this important in words: The areas of two parallelograms are to each other as the

products of the numerical values of their bases and atitudes. differente that the quotient ( would have a meaning only if a and If especially the second parallelogram is the unit square, i.e. a

square on the unit of length, then a' - =1, A'=u, and we have to have a common measure, until we introduce incommensurable numbers, while the ratio a : has always a meaning, and thus gives

À=or AaB.u. rise to the introduction of incommensurable

numbers. Thus it is really the theory of ratios in the fifth book which enables

This gives the theorem: The number of unit squares contained in us to extend the geometrical calculus given before in connexion

a parallelogram equals the product of the numerical values of base with Book II. It will also be seen that if we write the ratios in and altitude, and similarly the number of unit squares contained in Book V. as quotients, or rather as fractions, then most of the theorems a triangle cquals half the product of the numerical values of base state properties of quotients or of fractions. 864. Prop. 17. I} three straight lines are proportional the rectangle equal to the product of the base and the altitude, meaning by this

This is often stated by saying that the area of a parallelogram is contained by the extremes is equal to the square on the mean and product the product of the numerical values, and not the product as conversely, is only a special case of 16. After the problem,

Prop: defined above in $ 20. 18, On a given straight line to describe a rectilineal figure similar and similarly situated to a given rectilineal figure, there follows another diagonals, such as are considered in Book I., 43., They

$ 68. Propositions 24 and 26. relate to parallelograma nout fundamental theorem: Prop. 19. Similar triangles are to one another in the duplicate

Prop. 24. Parallelug, ams cbout the diameter of any surallelogram are similar

to the whole parallelogram and to one another; and its ratio of their homologous sides. In other words, the areas of similar triangles are to one another as the squares on homologous sides. angle, and be similarly situated, they are about ihe same diameter.

converse (Prop. 26), If two similar parallelograms have a common This is generalized in:

Between these is inserted a problem. Prop, 20. Similar polygons may be divided into the same number of similar triangles, having the same ratio to one another that the

Prop. 25. To describe a rectilineal figure which shall be similar to polygons have; and the polygons are to one another in the duplicate one given rectilinear figure, and equal to another given rectilineal ratio of their homologous sides. $65. Prop. 21. Rectilineal figures which are similar to the same

$ 69. Prop. 27. contains a theorem relating to the theory of

maxima and minima. We may state it thus: reciilineal figure are also similar to each other, is an immediate consequence of the definition of similar figures. As similar figures

Prop. 27. If a parallelogram be divided into two by a straight line may be said to be equal in " shape" but not in " size," we may state culling the bose, and if on half the base another parallelogram be con

strucled similar to one of those parts, then this third parallelogram is it also thus: “Figures which are equal in shape to a third are equal in shape greater than the other part.

Of far greater interest than this general theorem is a special case Prop. 22. ij four straight lines, be proportionals, the similar where one of the parts into which the parallelogram is divided is

of it, where the parallelograms are changed into rectangles, and rectilineal figures similarly described on ihem shall also be propor- made a square; for then the theorem changes into one which is tionals; and if the similar rectilineal figures similarly described on four easily recognized to be identical with the following: straight lines be proportionals, those straight lines shall be proportionals. This is essentially the same as the following:

of all rectangles which have the same perimeter the square hos the

greatest arta. a:b=c:d,

This may also be stated thus:-. then a : b2=c:d.

of all rectangles which have the same area the square has the least $66. Now follows a proposition which has been much discussed


$ 70. The next three propositions contain problems which may with regard to Euclid's exact meaning in saying that a ratio is

be said to be solutions of quadratic equations. The first two are, compounded of two other ratios, viz. : Prop. 23. Parallelograms which are equiangular to one another, scribe them as follows:

like the last, involved in somewhat obscure language. We tranhave to one another the ratio which is compounded of the ratios

of their

Problem.-To describe on a given base a parallelogram, and to sides. The proof of the proposition makes its meaning clear. In symbols a point on the base into two parallelograms, of which the one has

divide it either internally (Prop. 28) or externally. (Prop. 29) from the ratio a : c is compounded of the two ratios a :b and b:c, and if

a given size (is equal in area to a given figure), whilst the other a: b=a':0, 6:c=b":6", then a:c is compounded of a': 'b' and

has a given shape (is similar to a given parallelogram). fie'. If we consider the ratios as numbers, we may say that the one

If we express this again in symbols, calling the given base , the ratio is the product of those of which it is compounded, or in symbols, first case from the equations

one part x, and the altitude y, we have to determine x and y in the a abajo b_b.

(2-x)y=k, The theorem in Prop. 23 is the foundation of all mensuration of areas. From it we see at once that two rectangles have the ratio k? being the given size of the first, and p and the base and altitude of their areas compounded of the ratios of their sides.

of the parallelogram which determine the shape of the second of the II A is the area of a rectangle

contained by a and 6, and B that required parallelograms. of a rectangle contained by c and d, so that Arab, B=cd, then If we substitute the value of y, we get A: B=ab:

cd, and this is the theorem says, compounded of the ratios @ :c and 6:4. In forms of quotients,

(a -X)x


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ax-xels, This shows how to multiply quotients in our geometrical calculus, where a and b are known quantities, taking ba

Further, Two triangles have the ratios of their areas com pourided of the ratios of their bases and their altitude. For a triangle is equal in area to half a parallelogram which has the same base and the quadratic

The second case (Prop. 29) gives rise, in the same manner, to the sa me altitude. 8 67. To bring these theorems to the form in which they are usually

ax+* = 64. given, we assume a straight line u as our unit of length (generally

The next problem an inch, a foot, a mile, &c.),

and determine the number a which Prop. 30. To cut a given straight line in extreme and meon ratio, expresses how often u is contained in a line , so that a denotes the leads to the equation ratio a : " whether commensurable or not, and that a=au. Ve 1


This is, therefore, only a special case of the last, and is, besides, Prop. 6. Any two lines which are perpendicular lo the same plane an old acquaintance, being essentially the same problem as that are parallel to each other; and conversely proposed in II. 11.

Prop: 8. ll of two parallel straight lines one is perpendicular to e Prop. 30 may therefore be solved in two ways, either by aid of plane, the olher is so also. Prop. 29 or by aid of Il. 11. Euclid gives both solutions.

Prop. 7. If two straight lines are parallel, the straight line whisk 8.71. Prop. 31 (Theorem),, In any right-angled triangle, any joins any point in one to any point in the other is in the same plane as rectilineal figure described on the side subtending the right angle is the parallels. (See above, $ 73.), equal to the similar and similarly-described figures on the sides con Prop. 9: Two straight lines which are each of them parallel to the taining the right angle, -is a pretty generalization of the theorem of same straight line, and not in the same plane with it, are parallel to Pythagoras (1. 47).

one another; where the words, "and not in the same plane with 'Leaving out the next proposition, which is of little interest, we it," may be omitted, for they exclude the case of three parallels come to the last in this book.

in a plane, which has been proved before; and Prop: 33. In equal circles angles, whether at the centres or the Prop. 10. If two angles in different planes have the two limits of circumferences, have the same ratio which the arcs on which they stond the one parallel to those of the other, then the angles are equal. That have to one another; so also have the sectors.

their planes are parallel is shown later on in Prop. 15. of this, the part relating to angles at the centre is of special This theorem is not necessarily true, for the angles in question importance; it enables us to measure angles by arcs.

may be supplementary; but then the one angle will be equal to With this closes that part of the Elements which is devoted to that which is adjacent and supplementary to the other, and this the study of figures in a plane.

latter angle will also have its limits parallel to those of the first.

From this theorem it follows that if we take any two straight Book XI.

lines in space which do not meet, and if we draw through any point $72. In this book figures are considered which are not confined P in space two lines parallel to them, then the angle included by to a plane, viz. first relations between lines and planes in .space, these lines will always be the same, whatever the position of the and afterwards properties of solids.

point P may be. This angle has in modern times been called the Of new definitions we mention those which relate to the perpen- angle between the given lines: dicularity and the inclination of lines and planes.

By the angles between two not intersecting lines we understand the Def. 3. A straight line is perpendicular, or al righ! angles, lo : angles which two intersecting lines include that are parallel respectively plane when it makes right angles with every straighi line meeting it to the two given lines. in that plane.

$76. It is now possible to solve the following two problems: The definition of perpendicular planes (Def. 4) offers no difficulty. To draw a straight line perpendicular lo a given plane from a given Euclid di anes the inclination of lines to planes and of planes to point which lies planes (Ver 5 and 6) by aid of plane angles, included by straight 1, Nol in the plane (Prop. 11). lines, with which we have been made familiar in the first books.

2. In the plane (Prop. 12). The other important definitions are those of parallel planes, The second case is easily reduced to the first-viz. if by aid of which never meet (Def. 8), and of solid angles formed by three or the first we have drawn any perpendicular to the plane from some more planes meeting in a point (Def. 9):

point without it, we need only draw through the given point in the To these we add the definition of a line parallel to a plane as a plane a line parallel to it, in order to have the required perpendicular line which does not meet the plane.

given. The solution of the first part is of interest in itself. It de $. 73. Before we investigate the contents of Book XI., it will be pends upon a construction which may be expressed as a theorem. well to recapitulate shortly what we know of planes and lines from If froin a point A without a plane a perpendicular AB be drawn to the the definitions and axioms of the first book. There a plane has plane, and if from the foot B of this perpendicular another perpendicular been defined as a surface which has the property that every straight BC be drawn to any straighi line in the plane, then the straight line line which joins two points in it lies altogether in it.. This is equi- joining A to the foot of this second perpendicular will elso be perpenvalent to saying that a straight line which has two points in a plane dicular to the line in the plane. has all points in the plane. Hence, a straight line which does not The theory of perpendiculars to a plane is concluded by the lie in the plane cannot have more than one point in common with theoremthe plane. This is virtually the same as Euclid's Prop. I, viz. : Prop. 13. Through any point in space, whether in or without a

Prop. 1. One part of a straight line cannot be in a plane and another plane, only one straight line can be drawn perpendicular to the plane. part without it.

$77. The next four propositions treat oi parallel nes. It is It also follows, as was pointed out in $ 3, in discussing the defini- shown that planes which have a common perpendicular are paralld tions of Book I., that a plane is determined already by one straight (Prop. 14); that two planes are parallel if two intersecling straight line and a point without it, viz. if all lines be drawn through the lines in ihe one are parallel respectively to two straight lines in the point, and cutting the line, they will form a plane.

other plane (Prop: 15); that parallel planes are cut by any plane in This may be stated thus:

parallel straight lines (Prop. 16); and lastly, that any two straigk! A plane is determined

lines are cut proportionally by a series of parallel planes (Prop: 17). ist, By a straight line and a point which does not lie on il;

This theory is made more complete by adding the following 2nd. By three points which do not lie in a straight line; for if two theorems, which are casy deductions from the last: Two parallei of these points be joined by a straight line we have case 1; planes have common, perpendiculars (converse to 14); and Two

3rd, By two intersecting straight lines; for the point of interaction planes which are parallel to a third plane are parallel to each other. and two other points, one in each line, give case 2;

It will be noted that Prop. 15 at once allows of the solution of 4th, By two parallel lines (Def. 35, 1.).

the problem: “Through a given point to draw a plane parallel to The third case of this theorcm is Euclid's

a given plane.". And it is also easily proved that this problem Prop. 2. Two straight lines which cut one another ore in one plane, allows always of one, and only of one, solution. and three straight lines which meet one another are in one plane.

$78. We come now to planes which are perpendicular to one And the fourth is Euclid's

another. Two theorems relate to them. Prop. 7. If two straight lines be parallel, the straight line drawn Prop. 18. If a straight line be at right angles to e plane, every from any point in one io any point in the other is in the same plane plane which passes through it shall be at right angles to that plane. with the parallels. From the definition of a plane further follows. Prop. 19. If two planes which cut one another be each of them

Prop. 3. If two planes cut one another, their common section is a perpendicular to a third plane, their common section shall be per straight line.

pendicular to the same plane. 74. Whilst these propositions are virtually contained in the 879. If three planes pass through a common point, and if they definition of a plane, the next gives us a new and fundamental bound each other, a solid angle of three faces, or a lrihedral angle. property of space, showing at the same time that it is possible to is formed, and similarly by more planes a solid angle of more faces, have a straight line perpendicular to a plane, according to Def. 3. or a polyhedral angle. These have many properties which are quite It states

analogous to those of triangles and polygons in a plane. Euclid Prop. 4. IL , straight line is perpendicular lo iwo straight lines states some, viz., in a plane which it meels, then it is perpendicular lo all lines in ihe plane Prop. 20. If a solid angle be contained by three plane angles, cry which it meets, and hence it is perpendicular to the plane.

two of them are together greater than the third. Def, 3 may be stated thus: If a straight line is perpendicular But the next to a plane, then it is perpendicular to every line in the plane which Prop. 21. Every solid angle is contained by plane angles, which it meets. The converse to this would be

are together less than four right angles-has no analogous theorem AU straight lines which meet a given straight line in the same point, in the

plane. and are perpendicular to it, lie in a plane which is perpendicular lo We may, mention, however, that the theorems about triangles that line.

contained in the propositions of Book I., which do not depend This Euclid states thus:

upon the theory of parallels (that is all up to Prop. 27), have their Prop. 5. If three straight lines meet all at one point, and a straight corresponding theorems about tribedral angles. The latter are line stands at right angles to each of them at that point, the three straight formed, if for “ side of a triangle" we write plane angle lines shall be in one and the same plane.

"face" of trihedral angle, and for " angle of triangle we sub75. There follow theorems relating to the theory of parallel stitute " angle between iwo faces " where the planes containing the lines in space, viz.

solid angle are called its faces. We get, for instance, from I. 4. the


theorem, If two trihedral angles have the angles of two faces in the one which takes the place of the rectangle in the plane. If this has all equal to the angles of two faces in the other, and have likewise the angles its edges equal we obtain the “cube. induded by these faces equal, then the angles in the remaining faces are If we take a certain line u as unit length, then the square on u is equal, and the angles between the other faces are equal each to each, viz. the unit of area, and the cube on u the unit

of volume, that is to those which are opposite equal faces. The solid angles themselves are say, if we wish to mcasure a volume we have to determine how aot necessarily equal, for they may be only symmetrical like the many unit cubes it contains. right hand and the left.

A rectangular parallelepiped has, as a rule, the three edges unThe connexion indicated between triangles and trihedral angles equal, which meet at a point. Every other edge is equal to one will also be recognized in

of them. If a, b, c be the three edges meeting at a point, then we Prop. 22. If every two of three plane angles be greater than the may take the rectangle

contained by two of them, say by band third, and if the straight lines which contain them be all equal, a triangle as base and the third as altitude. Let V be its volume, V' that of may be made of the straight lines that join the extremilies of those equal another rectangular parallelepiped which has the edges a, b, c, straight lines.

hence the same base as the first. It follows then easily, from Prop. And Prop. 23 solves the problem, To construct a frihedral angle 25 or 32, that V:V'=a:a'; or in words, kaving the angles of its faces equal to three given planc angles, any two Rectangular parallelepipeds on equal bases are proportional to their of them being greater than the third. It is, of course, analogous to the altitudes. problem of constructing a triangle having its sides of given length. If we have two rectangular parallelepipeds, of which the first has Two other theorems of this kind are added by Simson in his the volume V and the edges a, b, c, and the second, the

volume V edition of Euclid's Elements.

and the edges a', b', c', we may compare them by aid of two new $ 80. These are the principal properties of lines and planes in ones which have respectively

the edges d', b, c and c', 6', 6, and the space, but before we go on to their applications it will be well to volumes Vi and V,.' We then have define the word distance. In geometry distance means always "shortest distance "; viz. the distance of a point from a straight

V:V,=:a'; V,:V,=6:8, V,:V'=c:d. line, or from a plane, is the length of the perpendicular from the Compounding these, we have point to the line or plane. The distance between two non-intersect

V:V'=(a: 2" (6:09 (0:0). ing lines is the length of their common perpendicular, there being or but one. The distance between two parallel lines or between two

6 parallel planes is the length of the common perpendicular between

Vthe lines or the planes.

$ 81. Parallelepipeds. —The rest of the book is devoted to the Hence, as a special case, making V' equal to the unit cube U on u study of the parallelepiped. In Prop. 24 the possibility of such we get a solid is proved, viz.

Vabc Prop. 24. If a solid be contained by six planes two and two of

Uu.' *

= a.B. 7, which are parallel, the opposite planes are similar and equal parallel- where a; B. y are the numerical values of a. 8, c; that is, The number gamis, Euclid calls this solid henceforth a parallelepiped, though he of unit cubes in a rectangular parallelepi pod is equal to the product

of the numerical values of its three edges. This is generally exnever defines the word. Either face of it may be taken as base, pressed by saying the volume of a rectangular parallelepiped is and its distance from the opposite face as bltitude.

measured by the product of its sides, or by the product of its base Prop. 25. If a solid parallelepiped be cut by a plane parallel to into its altitude, which in this case is the same. two of ils opposite planes, it divides the whole into two solids, the base of one of which shall be to the base of the other as the one solid is to the 28 or 40, to triangular prisms.

Prop. 31 allows us to extend this to any parallelepipeds, and Props. other. This theorem corresponds to the theorem (VI. 1) that parallelo-measured by the product of base and altitude.

The volume of any parallelepiped, or of any iriangular prism, is grams between the same parallels are to one another as their bases. A similar analogy is to be observed among a number of the remaining a number of triangular prisms, which have the same altitude and

The consideration that any polygonal prism may be divided into propositions.

the sum of their bases equal to the base of the polygonal prism, 82. After solving a few problems we come to

shows further that the same holds for any prism whatever, Prop. 28. If a solid parallelepiped be cut by a plane passing through the diagonals of two of the opposite planes, it shall be cut in

Book XII. treo equal parts. In the proof of this, as of several other propositions, Euclid

$ 85. In the last part of Book XI. we have learnt how to compare neglects the difference between solids which are symmetrical like

the volumes of parallelepipeds and of prisms. In order to determine the right hand and the left.

the volume of any solid bounded by plane faces we must determine Prop. 31. Solid parallelepipeds, which are upon equal bases, and into a number of pyramids.

the volume of pyramids, for every such solid may be decomposed of the same altitude, are equal to one another. Props. 29 and 30 contain special cases of this theorem leading up pyramids, it becomes only necessary to determine their volume.

As every pyramid may again be decomposed into triangular to the proof of the general theorem.

This is done by the
As consequences of this fundamental theorem we get
Prop. 32. Solid parallelepipeds, which have the same allitude, are

Theorem.-Every triangular pyramid is equal in volume to one to one another as their bases; and

third of a triangular prism having the same base and the same

altitude as the pyramid. Prop. 33. Similar solid parallelepipeds are to one another in the triplicate ratio of their homologous sides.

This is an immediate consequence of Euclid's If we consider, as in $67, the ratios of lines as numbers, we may

Prop. 7. Every, prism having a triangular base may be divided

into three pyramids that have triangular bas and are equal to one also say

another. The ratio of the volumes of similar parallelepipeds is equal to the ratio of the third powers of hoinologous sides.

The proof of this theorem is difficult, because the three triangular Parallelepipeds which are not similar but equal arc compared by shape, and cannot be made to coincide. It has first to be proved

pyramids into which the prism is divided are by no means equal in aid of the theoren Prop. 34. The bases and altitudes of equal solid parallelepipeds equal bases and cqual'altitudes, This Euclid does in the following

that two triangular pyramids have cqual volumes, if they have and reciprocally proportional; and if the bases and altitudes be reciprocally proportional, the solid parallelepipeds are equal.

He first shows, (Prop: 3) that a triangular pyramid may

be divided into four parts, of which two are equal triangular pyramids $ 83. Of the following propositions the 37th and 4oth are of

similar to the whole pyramid, whilst the other two are equal trispecial interest.

Prop: 37. If four straight lines be proportionals, the similar solid angular prisms, and further, that these two prisms together are parallelepipeds, similarly described from them, shall also be pro- greater than the two pyramids, hence more than half the given portionals; and if the similar parallelepipeds similarly described pyramid. He next shows (Prop. 4) that if two triangular pyramids from four straighi lines de proportionals, the straight lines shall be are given, having equal bases and equal altitudes, and it each be

divided as above, then the two triangular prisms in the one are proportionals. In symbols it says

equal to those in the other, and each of the remaining pyramids in

the one has its base and altitude equal to the base and altitude of If a:b=c:d, then a':03=co:d'.

the remaining pyramids in the other. Hence to these pyramids the Prop. 40 teaches how to compare the volumes of triangular same process is again applicable. We are thus enabled to cut out prisms with those of parallelepipeds, by proving that a triangular of the two given pyramids equal parts, each greater than half the prism is equal in volume to parallelepiped, which has its altitude original pyramid. of the remainder we can

again cut out equal and its base equal to the altitude and ihe base of the triangular parts greater than half these remainders, and so on as far as we like. prism.

This process may be continued till the last remainder is smaller $84. From these propositions follow all results relating to the than any assignable quantity, however small. It follows, so we mensuration of volumes.' We shall state these as we did in the case should conclude at present, that the two volumes must be equal, for of areas. The starting point is the “rectangular" parallelepiped, they cannot differ by any assignable quantity. which has every edge perpendicular to the planes it meets, and To Greek mathematicians this conclusion offers far greater


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