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This theorem may also be stated thus :

8 36. Propositions 30, 33, 34, contain problems which are solved Through three points only one circumference may be drawn; or, by aid of the propositions preceding them :Three points determine a circle.

Prop. 30. To bisect a given arc, that is, to divide it into two equal Euclid does not give the theorem in this form. He proves, how. parts ; ever, that the two circles cannot cut another in anore than two points Prop. 33. On a given straight line to describe a segment of a circle (Prop. 10), and that two circles cannot touch one another in more containing an angle equal to a given rectilineal angle ; points than one (Prop. 13).

Prop. 34. From a given circle to cut off a segment containing an $ 30. Propositions 11 and 12 assert that if two circles touch, then angle equal to a given rectilineal angle. the point of contact lies on the lino joining their centres. This gives 8 37. If we draw chords through a point A within a circle, they two propositions, because the circles may touch either internally will each be divided by A into two segments. Between these or externally.

segments the law holds that the rectangle contained by them has § 31. Propositions 14 and 15 relate to the length of chords. The the same area on whatever chord through A the segments are first says ; that equal chords are equidistant from the centre and that taken. The value of this rectangle changes, of course, with the chords which are equidistant from the centre are equal ;

position of A. Whilst Prop. 15 compares unequal chords, viz., Of all chords the A similar theorem holds if the point A be taken without the diameter is the greatest, and of other chords that is the grcater which circle. On every straight line through A, which cuts the circle is nearer to the centre ; and conversely, the greater chord is nearer to in two points B and C, we have two segments AB and AC, the centre.

and the rectangles contained by them are again equal to one $32. In Prop. 16 the tangent to a circle is for the first time in another, and equal to the square on a tangent drawn from A to troduced. The proposition is meant to show that the straight line the circle. at the end point of the diameter, and at right angles to it is a tan. The first of these theorems gives Prop. 35, and the second Prop. gent. The proposition itself does not state this. It runs thus - 36, with its corollary, whilst Prop. 37, the last of Book III., gives

Prop. 16. The straight line drawn at right angles to the diameter the converse to Prop. 36. The first two theorems may be combined of a circle, from the extremity of it, falls without the circle ; and no

in one : straight line can be drawn froin the extremnity, belween that straight THEOREM.-If through a point A in the plano of a circle a straight line and the circumference, 80 as not to cut the circle.

line be drawn cutting the circle in B and C, then the rectangle Corollary.— The straight line at right angles to a diameter drawn AB. AC has a constant value so long as the point A be fixed ; and if through the end point of it touches the circle.

from A a tangent AD can be drawn to the circle, touching at D, then The statement of the proposition and its whole treatment show the above rectangle cquals the square on AD. the difficulties which the tangents presented to Euclid.

Prop. 37 may be stated thus :Prop. 17 solves the problem through a given point, either in the THEOREM.-If from a point A without a circle a line be drawn cutcircumference or without it, to draw a tangent to a given circle. ting the circle in B and C, and another line to a point D on the

Closely connected with Prop. 16 are Propg. 18 and 19, which circle, and if AB. AC - AD, then the lino AD touches the circle state (Prop. 18), that the linc joining the contre of a circle to the point at D. of contact of a tangent is perpendicular to the tangent; and con- It is not difficult to prove also the converse to the general proversely (Prop. 19), that the straight line through the point of contact position as above stated. It may be expressed as follows :of, and perpendicular to, a tangent to a circle passes through the centro If four points ABCD be taken on the circumference of a circle, and of the circle.

if the lines AB, CD, produced if necessary, meet at E, then $ 33. The rest of the book relates to angles connected with a

EA. EB

EC. ED; circle, viz., angles which have the vertex either at the centre or on the circumference, and which are called respectively angles at

and concerscly, if this relation holds then the four points lie on a the centre and angles at the circumference. Between these two

circle, that is, the circle drawn through three of them passes through kinds of angles exists the important relation expressed as

the fourth. follows :

That a circle may always be drawn through three points, proProp. 20. The angle at the centre of a circle is double of the angle vided that they do not lie in a straight line," is proved only later at the circumference on the same base, that is, on the same arc.

on in Book IV. This is of great importance for its consequences, of which the two following are the principal :

Book IV. Prop. 21. The angles in the same segment of a circle are equal to one another;

§ 38. The fourth book contains only problems, all relating to And Prop. 22. The opposite angles of any, quadrilateral figure the construction of triangles and polygons inscribed in and circum, inscribed in a circle are together equal to two right angles.

scribed about circles, and of circles inscribed in or circumscribed Further consequences are :

about triangles and polygons. They are nearly all given for their Prop. 23. On the same straight line, and on the same side of it, own sake, and not for future use in the construction of figures, there cannot be tro similar segments of circles, not coinciding with as are most of those in the former books. In seven definitions one another ;

at the beginning of the book it is explained what is `nderstood And Prop. 24. Similar segments of circles on equal straight lines by figures inscribed in or described about other figures, with are equal to one another.

special reference to the case where one figure is a circle. Instead, The problem Prop. 25, A segment of a circle being given to however, of saying that one figure is described about another, it is describe the circle of which it is a segment, may be solved much more now generally said that the one figure is circumscribed about the easily by aid of the construction described in relation to Prop. 1, other. We may then state the definitions 3 or 4 thus III., in 8 27.

Definition. -A polygon is said to be inscribed in a circle, and the 8 34. There follow four theorems connecting the angles at the circle is said to be circumscribed about the polygon, if the vertices centre, the arcs into which they divide the circumference, and the of the polygon lie in the circumference of the circle. chords subtending these arcs. They are expressed for angles, arcs, And definitions 5 and 6 thus : and chords in equal circles, but they hold also for angles, arcs, and Definition. - A polygon is said to be circumscribed about a circle, chords in the same circle.

and a circle is said to be inscribed in a polygon, if the sides of the The theorems are :

polygon are tangents to the circle. Prop. 26. In equal circles equal angles stand on equal arcs, whether § 39. The first problem is merely constructive. It requires to they be at the centres or circumferences ;

draw in a given circle a chord equal to a given straight line, which Prop. 27 (converse to Prop. 26). In equal circles the angles which is not greater than the diameter of the circle. The problem is not stand on equal arcs are equal to one another, whether they be at the a determinate one, inasmuch as the chord may be drawn from any centres or the circumferences;

point in the circumference. This may be said of almost all proProp. 28. In equal circles equal straight lines (equal chords) cut | blems in this book, especially of the next two. They are : off equal arcs, the greater equal to the greater, and the less equal to Prop. 2. In a given circle to inscribe a trianglo equiangular to a the less ;

given triangle ; Prop 29 (converse to Prop. 28). In equal circles equal arcs are Prop. 3. About a given circle to circumscribe a triangle equi sublended by equal straight lines.

angular to a given triangle. & 35. Other important consequences of Props. 20-22 are :- Š 40. Of somewhat greater interest are the next problems, where

Prop. 31. In a circle the angle in a semicircle is a right angle ; | the triangles are given and the circles to be found. bul the angle in a segment greater than a semicircle is less than a Prop. 4. To inscribe a circle in a given triangle. right angle; and the angle in a segment less than a semicircle is The result is that the problem has always a solution, viz., the greater than a right angle ;

centre of the circle is the point where the bisectors of two of the Prop. 32. If a straight line touch a circle, and froin the point of nterior angles of the triangle meet. The solution shows, though contact a straight line be drawn cutting the circle

, the angles which Euclid does not state this, that the problem has but one soluthis line makes with the line touching the circle shall be equal to the tion; and also, angles which are in the alternate segments of the circle.

THEOREM. - The three bisectors of the interior angles of any triangle

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32

meet in a point, and this is the centre of the circle inscribed in the sides of any regular polygon mect in a point. The straight lincs triangle.

bisecting the ongles in the nigular polygon meet in the su me point. The solutions of most of the other problems contain also theorems. This point is the centre of the circles circumscribed about and Of these we shall state those which are of special interest ; Luclid inscribed in the regular polygon. Tho proof, which is easy, is left does not state any one of them.

to the reader. § 41. Prop. 5. To circumscribe a circle about a given triangıc. We can lisect any given arc (Prop. 30, III.). Hence we can Tho one solution which always exists contains the following :-- divide a circumference into an equal parts as soon as it has beca

TIIEOREM.-- The threr straight lincs which bisect the sides of a divided into n equal parts, or as soon as a regular polygon of a triangle at right angles meet in a point, and this point is the centre sides has been constructed. Henceof the circle circumscribed about the triangle.

THEOREM. ---If a regular polygon of n sules nuls burn conslructal, Euclid adds in a corollary the following property :

then a regular polygon of 2n sides, of 4n, of 8n sirles, dr., may The centre of the circle circumscribed about a triangle lics also be construción. Euclid shows how to construct regular poly. within, on a side of, or without the triangle, according as tho gons of 3, 4, 5, and 15 sides. It follows that we can construct triangle is acuto-angled, right-angled, or obtnse-angled.

regular polygons of § 42. Whilst it is always possible to draw a circle which is

0, 12,

sides inscribed in or circumscribed about a given triangle, this is not the

o 16, case with quadrilaterals or polygons of more sides. Of those for

5, 10

20, which this is possible the regular polygons are the most interesting.

15, 30, 60, 20. In each of them a circle may be inscribed, and another may be circumseribed about it.

The construction of any new regular polygon not included in one Euclid does not use the word regular, but he describes the

of these series will give rise to a new series. Till the beginning polygons in question as equiangular and cquilatcral. We shall

of this century nothing was added to the knowledge of regular ase the name regular polygon. The regular triangle is equi- polygons as given by Euclid. Then Gauss, in his celebrated lateral, tho regular quadrilateral is the square,

Arithmetic, proved that every regular bolgon of 2* +1 sides Euclid considers the regular polygons of 4, 5, 6, and 15 sides.

mny be constructed if this number 2 +1 be prime, and that no For each of the first three ho solves the problems-(1) to inscube

others can be constructed by elementary methods. This show such a polygon in a given circle; (2) to circumscribe it about i that regular polygons of 7, 9, 13 sides cannot thus be constructed, given circle ; (3) to inscribe a circle in, and (4) to circumscribe

but that a regular polygon of 17 sides is possible ; for 17 - 2 +1. a circle about, such a polygom

The next polygon is one of 257 sides. The coustruction becomes For the regular triangle the problems are not repeated, because already rather complicated for 17 sides. moro general problems have been solved Preps. 6, 7, 8, and 9 solve these problems for the square.

BOOK V. The general problem of inscribing in a given circle a regular polygon of n sides depends upon the problem of dividing the cir- & 47. The fifth book of the Elements is no exclusively geomecumference of a circle into n equal parts, or what comes to the same trical. It contains the theory of ratios and proportion of quantities thing, of drawing from the centre of the cirele n radii such that tho

in general. The treatment, as here given, is admirable, and angles betwoon consocutive radii are equal, that is, to divide the space about the centre into n equal angles. Thus, if it is required Euclid's theory is now generally replaced. It has, however, the

in every respect superior to the algebraical method by which to inscribe a square in a circle, we have to draw four lines from the

reputation of being too difficult for schools, and is therefore very centre, making the four angles equal This is done by drawing seldom read. We shall try to make the subject clear, and to show two diameters at right angles to one another. The ends of these why the usual algebraical treatment of proportion is not really diameters aro the vertices of the required square. lf, on the other sound. We begin by quoting those definitions at the beginning of hand, tangents be drawn at these ends, we obtain a square circum

Book V. which are most important. These definitions have given scribed about the circle.

rise to much discussion. & 43. To construct a regular pentagon, we find it convenient first

The only definitions which are essential for the fifth book are to construct a regular decagon. This requires to divide the space | Defs. 1, 2, 4, 5, 6, and 7. Of the remainder 3, 8, and 9 are more about the centre into ten equal angles. Each will be oth of a right than ustless, and probably not Euclid's, but additions of later angle, or }th of two right angles. If we suppose the decagon con

editors, of whom Theon of Alexandria was the most prominent. struotes, and if we join the centre to the end of one side, we get an Defs. 10 and u belong rather to the sixth book, whilst all the iyoceles triangle, where the angle at the centre equals 4th of two others are ucrely uominal. The really important outs are 4, 5, right angles ; hence each of the angles at the base will be {ths of

o, and 7. two right angles, as all three angles together equal two right angles. § 48. To define a magnitude is not attempted by Euclid. The Thus we have to construct an isoceles triangle, having the angle at first two definitions state what is meant by a “part,” that is, & the vertex cqual tv half an angle at the base. This is solved in submultiple or measure, and by a “multiple” of a given magnil'rop. 19, by aid of the problem in Prop. 11 of the second book. If tude. The meaning of Dct. 4 is that two given quantities can have we make the sides of this triangle equal to the radius of the gireu circle, then the base will be the side of the regular decagon their magnitude, that is, if they are of the same kind.

a ratio to one another only in case that they are comparable as to inscribud in the circle. This side being known the decagon can be 1 Dei. 3, which is probably due to Theon, professes to define a constructed, and if the vertices are joined alternately, leaving out ratio, but is as meaningless as it is uncalled for, for all that is half their number, we obtain the regular pentagon.

wanted is given in Deis. 5 and 7. Euclid does not proceed thus. He wants the pentagon before

In Def. 5 it is explained what is meant by saying that two the decayon. This, however, does not change the real nature of magnitudes have the same ratio to one another as two other his solr:tion, nor does his solution become simpler by not mention. magnitudes, and in Def. 7 what we have to understand by a greater ing the decagon. Once the regular pentagon is inscribed, it is easy to circumscribe meaning of the word proportional.

or a less ratio. The 6th definition is only nominal, explaining the anothci by drawing taugents at the vertices of the inscribed pen

Euclid represents magnitudes by lines, and often denotes them tigon. This is shown in Prop. 12.

either by single letters or, like lines, by two letters. We shall use Prop. 13 and 14 teach how a circle may be inscribed in or cir. only single letters for the purpose. If a and b denote two magni cumscribed about any given regular pentagon.

tudes of the same kind, their ratio will be denoted by a :); if $ 44. The regulur hexagon is more easily constructed, as shown in Prop. 15. The result is that the side of the regular hexagon

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c and d are 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 inscribed in a circle is equal to the radius of the circle.

ratio to one another as a and b, this will be expressed by writingFor this polygon the other three problems mentioned are not solved.

a :6::C:d. § 45. The book closes with Prop. 16. To inscribe a reguar Further, if m 18 a (whole) number, ma shall denote the multiple quindecagon in a given circle. That this may be done is ensily of a which is obtained by taking it in times.

If we inscribe a regular pentagon and a regular hexagon in § 49. The whole theory of ratios is based on Def. 5. the circle, having one vertex in cominon, then the arc from the Def. 5. The first of four magnitudes is said to have the same ratio common vertex to the next vertex of the pentagon is {th of the cir- to the sccond that the third has to the fourth when, any equimultiples cumference, and to the next vertex of the hexagon is 4th of the whatever of the first and the third being taken, and any equimul circumferenco. The differenco between these arcs is, therefore, tiples whatever of the second and the fourth, if the multiple of the first *-4-goth of the circumference. The latter may, therefore, be be less than that of the second, the multiple of the third is also less divided into thirty, and hence also in fifteen equal parts. and the than that of the fourth; and if the multiple of the first is equal to regular quindecagou be described.

that of the sccond, the multiple of the third is also equal to that of § 46. We concludo with a few theorema nbout regular polygons the fourth ; and if the inultiple of the first is greater than that of which are not given by Euclid.

the second, the multiple of the third is also greater than that of

the THEOREN. — The straight lines piracicular to and Visccting the fourth.

seen.

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It will be well to show at once in an example how this definition, therefore if

mane, then mb>nc, can be used, by proving the first part of the first proposition in the if

ma == nc, then mb= ne, sixth book. Triangles of the same altitude are to one another as if

ma<ne, then mb< nc, txir bases, or if a and b are the bases, and a and B the areas, of two triangles which have the same altitude, then a :b::a:B.

therefore the first proportion holds by Definition 5. To prove this, we have, according to Definition 5, to shuw

Prop. 8. If

a>b, then a :c>b:c,

and if ma>b, then manB, ·

cia<c:6.
if ma - nb, then ma-

The proof depends on Definition 7.
nB,
if mac nb, then manB.

Prop. 9 (converse to Prop. 7). If
That this is true is in our case easily seen. We may suppose that

a:c::b:c, the triangles have a common vertex, and their bases in the same

or if

c:a::c:b, then a = b. line. We set off the base a along the line containing the bases Prop. 10 (conversc to Prop. 8). If m times; we then join the different parts of division to the vertex,

a :c>b:c, then a> and get m triangles all equal to a. The triangle on ma as base equals, and if

cia<c: 0, then aco, therefore, m. If we proceed in the same manner with the basu b, setting it off n times, we find that the area of tho triangle on the

Prop. 11. If

a:6::c:d,

and base ab equals nb, the vertex of all triangles bring the same.

a:b::e:f, But

then if two triangles have the same altitude, then their areas are equal

c:d ::6:f. if the buses are equal; hence monß if ma = nb, and if their bases

In words, if two ratios are equal to a third, they are equal to one are unequal, then that has the greater area which is on the greater right to consider a ratio as a magnitude, for only now can we

another. After these propositions have been proved, we have a k252; in other words, ma is greater than, equal to, or less than 8. B, according ag ma is greater than, equal to, or less ihan nb, which

consider a ratio as something for which the axiom about magniwas to be proved.

tudes holds : things which are equal to a third are equal to one $ 50. It will be seen that even in this example it does not

another. become evident what a ratio really is. It is still an open question

We shall indicate this by writing in future the sign

instead whether ratios are magnitudes which we can compare.

We do of ::. The remaining propositions, which explain theinselves, may not know whether the ratio of two lines is a magnitude of the

then be stated as follows: same kind as the ratio of two areas. Though we might say

$ 53. Prop. 12. If a:b-c:d-e:f, that Def. 5 defines equal ratios, still we do not know whether they

then

a+c+6:6+d+f-«: b. are equal in the sense of the axiom, that two things which are

Prop. 13. If

a : b-c:d and c:dc:f, equal to a third are equal to one another. That this is tho case then

«:b>e: f. requires a proof, and until this proof is given we shall use the Prop. 14. Ti a:b-c:d, and a>c, then b>d. :: instead of the sign -, which, however, we shall afterwards Prop. 15. Magnitudes hayo the same ratio to one another that introduce.

their equimultiples haveAs soon as it has been established that all ratios are like magni.

ma :mb:. tudas, it becomes easy to show that, in some cases at least, they

Prop. 16. If a, 0, c, d are magnitudes of the same kind, and if are numbers. This step was nover made by Greek mathematicians. They distinguishod always most carefully between continuous

a :-?:d,

then magnitudes and the discrete series of numbers. In modern times

Q :cb:d. it has become the custom to ignoro this disserence.

Prop. 17. If a +6:6-c+d:d, !f, in determining the ratio of two lines, a common measure can

then

Q:bme: d. be sound, which is contained in times in the first, and n times in. Prop. 18 (converso to 17). If the second, then tho ratio of the two lines equals the ratio of the

a : -c:d two numbers min. This is shown by Euclid in Prop. 5, X. But the then

a +3:b-c+d :d. ratio of two numbers is, as a rule, a fraction, and the Greeks did Prop. 19. If a, b, c, d aro quantities of the same kind, and if not, as we do, considor fractions as numbers: Far less had they

a:bc:d, any notion of introducing irrational numbers, which are neither

tlen

Q-C:b-d-a:b. whole nor fractional, as we are obliged to do if we wish to say that all ratios aro numbors. The incommensurable numbers which are

§ 54. Prop. 20. If there be three magnitudes, and other three, thus introduced as ratics of incommensurablo quantities are 101-a.

urhich have the same ratio, token two and two, then if the first be days as familiar to us as fractions ; but a proof is generally omitted greater than the third, the fourth shall be greater than the sixth; that we may apply to them the rules which havo been established and if cqual, cqual; and if less, less. for rational numbers only. Euclid's treatment of ratios avoids this

If we understand by difficulty. His definitions holds for commensurable as well as for

a : 6:c;d:e:...

d': :8:d::d:... incomnicnsurable quantities. Even tho notion of incommensurable that the ratio of any two consecutive magnitudes on the first side quantitios is avoided in Book V. But ho proves that tlo niore cquals that of the corresponding magnitudes on the second side, clementary rules of algebra hold for ratios. We shall stato all we may write this theorem in symbols, thus :his propositions in that algebraical form to which we are now If a, b, c be quantities of one, and d, e, f magnitucles of the saine accustomed. This may, of course, bo done without changing the or any other kind, such that character of Euclid's method.

a: b:c-die:f, $ 51. Using the notation explained above to express the first and if

a>c, then af, propositions as follows :

but if

a-c, then dof,

and if Prop. 1. If a-md', b - mb', c-me',

a<c, then di thien a+b+c-(a' + b + c).

Prop. 21. If

a': 5- - : f and b:c-d:6,

1 1 Prop. 2. If -mb, and c-md,

Q:6:=

a 6 -- nb, and f-nd,

and if

a>c, then >j, then a + c is the same multiple of b as c+f is of d, viz. :

but if

( C,

then des

and if ate - (on +92)6, and c+f-(m + nld.

a<c, then dc. Prop. 3. If r- mb, c- - ind, then is na the same multiple of b

By aid of theso two propositions the following two are proved. that nie is of d, viz., nu - nmb, nc nind.

8 55. Prop. 22. If there be any number of magnitules, and as

many others, which harc the same ratio, taken two and two in order, Prop. 4. If a:b::c:d,

the first shall have to the last of the first magnitudes the same ratio then ina : 110 :: mc : nd.

which the first of the others has to the last. Prop. 5. If a-mb, and c- ma

We may state it more generally, thus :thon - m(b-d).

If a : 6:c:d:c:... -a':8::d:d:..., Prop. 6. If @-mb, c- - md,

then not only havo tivo consecutive, but any two magnitudes on then are a – nb and c-nd either equal to, or equimultiples of, 8 the first sido, the same ratio as the corresponding magnitudes og and d, viz., a- nb-(m – 1236 and -nd-(m, n)d, where in - - the other. For instancemay be unity.

C:C-a : 6:6 = 6':', &c. All these propositions relate to equimultiples. Now follow propositions about ratios which are compared as to their magnitudo.

Prop. 23 we state only in symbois, riz. :

1 1 1 $ 52. Prop. 7. If a - b, then a :C::6:c and c:a::c:b.

If

a:0:0:0:6:

U = mbi The proof is simply this. As a - b we know that ma

Ö:2:7:2

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or if

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- C

:d ;

1

1

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a

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:

:

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then
a:c=:a',

Corollary.- From this it is manifest that the perpendirular 6:e-(':',

drawn from the right angle of a right-anglcu triangle to the base and so on.

is meal proportional between the segments of the base, and also Prop. 24 comes to this : Ifa.6-c:d and c:6-9:d, then thut etch of the sides is a mean proportional between the base and a+c:b-c+f:d.

the segment of the base adjacent to that side.

$ 61. There follow four propositions containing problems, riz, Some of the proportions which are considered in the above pro in language slightly different from Euclid's:positions have special names. These we have omitted, as being of Prop. 9. To divide a struiyht line into a girch number of quel uo use,

since algebra has enabled us to bring the different operations purts. contained in the propositions under a common point of view.

Prop. 10. To divide a straight line in a giren ratio. $ 56. The last proposition in the fifth book is of a different

Prop. 11. I'o find a third proportional to two yiron si raryht lince, character.

Prop. 12. To firul a fourth proportional to threc gioen smight Prop. 25. If four mugiitules of the same kind be proportionul, lines. the greatest and least of them together shall be grcater than uic other

Prop. 13. To finil al can proportional beturen tvo giren straight two together. In symbols

lincs. If a, b, c, d be magnitudes of the same kind, and if a :)=0:11,

The last three may be written as equations with one unknown and if a is the greatest, hence d the least, then a+b+c. $ 57. We return onco agnin to the question, Whnt is a ratio? required line 2, we have to find a line x so that

quantity,- viz., if we call the given straight lines a, b, g, and the We have seen that we may trcat ratios as innniindes, and that all ratios are magnitudes of the same kinıl, for we may coinpire any

Pror. 11.

: b = 1:7; two as to their magnitude. It will presently bo shown that ratios

Prop. 12.

a: b = 6:2; of lines may be considered as quotients of lines, so that a ratio

Prop. 13.

r:.:b. apreurs as answer to the question, How often is one line contained We shall see presantly how thesc may be written without the in another? But the answer to this question is given liy il number, signs of ratios, at least in some cases, and in all cases if we adinit incommensurable $ 62. Euclid considers next proportious connected with parallelonumbers. Considered from this point of view, we may say the fifth grams and trinngles which are equal in area. book of the Elements shows that some of the simpler algelornical Prop. 14. Equal parallelograms which lure une angle of the one operations hold for incommensurable nuinbers. In the ordinary quuliv one angle wilher oher hure their sides whout the qual angles algebraical treatment of numbers this proof is altogether omitted; urciprocally proportional: and parallelogrung which have one angle or given by a process of limits which does not scow to be natural of the one equil to one angic of the other, and their sides about the to the subject.

cqual amylrs reciprocally proportional, arr equal to one another.

l'rop. 15. Equal tringles which hare one angle of the one cqual

to one angle of the other, hard their sidrs about the cqunl unupirs Book 1).

reciprocally proportional; and trianglıs which hari oni angle of the $ 58. The sixth book contains the theory of similar figues,

one equal to one angle of the other, and their sides abmit the equal · After a few definitions explaining terms, the first proposition gives ongles reciprocally proportional, are equal to cn uroller. the first application of the theory of proportion.

The Intter proposition is really the saine as the former, for if, 99 Prop. 1. Triangles and parallclograms of the same altitude are

in the arcompanying to one another as thcir bases.

dlingram, in the figuro The proof has already been considered in $ 49.

Tulonging to the former From this follows easily the important theorom

the two equul jaralleloProp. 2. If a straight line be drawn parallel to one of the sidrs

grams AB and BC bu of a triangle, it shall cut the other sides, or those sides produced, pro

bisectrd by the lines DF

B portionally; and if the sides or the sides produced be cut proportion.

and EG, and if EF ho ally, the straight line which joins the points of sccion shall be

drawn, we get tlıc figure parallel to the remaining side of the triangle

belonging to the latter. $ 59. The next proposition, together with one wilded by Simson

It is worth noticing

that the lines FE and as Prop. A, may be expressed more conveniently if we introduce a inodern phrascology, viz., if in a line AB ire assume a point C

DG are parallel.

We between A and B, we shall say that C divides AB internally in the

may state therefore the tlicorcm-. ratio AC : CB ; but if C be taken in the line AB produced, we

I'HEOREM.--If tuo trianyles arc equal in arca, and have one angle shall say that AB is divided externally in the ratio AC CB.

in the onc rctically opposite to one angle in the other, then the teru The two propositions then come to this :

straight lines which juin the remaining tuo rertices of the one to those THEOREM (Prop. 3). --The lisector of an angle in a trinngic of the other triangle are parallel. divides the opposite side internally in a ratio cquni to the ratio of ihr $ 63. A most important theorcm is two sides including that angle; anıl conversely, if a line through thc

l'rop. 16. If four straiglu lincs br proportionals, the techngle vertex of a triangle diride the basc internally in the rrutio of the tiro

"contained by the extrc:nes is cqual to the rectangle contained by the other sidcs, then that line bisccts the aiule at the rcrlcx.

mcans ; and is the reclangle contained by the extreines be equal to the THEOREM (Simson's Prop. A). I'lc linc which lisccts

rectangle contained by the ineans, the four straight lines are proporcatcrior angle of a trinng" divides the opposite sido wlcrnally in the

tionais. ratio of the other sides, and conversely, if a line through the vortex

In synıbols, if a, 1, c, d are the four lincs, and of a triangle divide the brise externally in the ratio of the sides, then if

a:b-c:d, it bisccis an erterior angle at the rerter of the trianglc.

then

ad bc ; If we coinbine both we have

and conversely, if

ad - bc, THEOREM. ---The trvo lines which biscct the interior and crterior then

a:bc:d, angles at one vcrtez: of a triangle divide the opposite side internally

where. ad and bc denote (as in $ 20), the areas of the rectangles and externully in the same ratio, riz., in the ratio of the other tuo

contained by a and d and by b and c respectively. sides. $ 60. The next four propositions contain the theory of similar.

This alloirs us to transform every proportion between four lines triangles, of which four cases are considered. They may be stated

into an equation between two products.

It shows further that the operation of forming a product of two together THEOIEN.— Two triangles are similar,

lines, and the operation of forining their ratio are each the inverse

of the other. 1. (Prop. 4). If the triangles are equiangular;

2. (Prop. 5). If the sides of the one are proportional to those of If we now define a quotient of two lines as the number which the other ;

3. (Prop. 6). If two sides in one are proportional to two sides in multiplied into o gives a, so that the other, and if the ongles contained by thesc sides are eqrial;

4. (Prop. 7). If two sides in one are proportional to two sides in the other, if the angles opposite homologous sides are equal, and if the angles opposite the other homologous sides are both acute, both right,

we see that from the equality of two quotients or both obtuse; homologous sides being in each case those which are

a opposite equal angles.

d An important application of these theorems is at once made to a right-angled triangle, viz.. :

follows, if we multiply both sides by bd, Prop. 8. In a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another,

E

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But from this it follows according to the last theorem that If now a, 6 be base and altitude of one, a', ' those of another a :b - c:d.

parallelogram, a, B and a', s' their numerical values respectively, Hence we conclude that the quotient

and A, A' their areas, then
and the ratio a :b are
8

A

6 different forms of the same magnitude, ouly with this important

A 7 à a B difference that the quotient

6

would have a meaning only if a and In tords : The arcas of tico parallcloyrams are to cach other as the o have a common measure, until we introduce incommensurable products of the numerical values of their bases and altitudes.". numbers, while the ratio a : 0 has always a meaning, and thus gives

If especially the second parallelogram is the unit sajuare, i.c., a rise to the introrluction of incommensurable numbers.

square on the unit of length, then á B - 1, A' 11", and we

have Thus it is really the theory of ratios in the fifth book which enables us to extend the geometrical calculus given before in con.

=aß. or A

. u. nexion with Book II. It will also be seen that if we write the ratios in Book V. as quotients, or rather as fractions, then most of This gives the theorem: The number of unit squares contained in the theorems state properties of quotients or of fractions.

a parallelogram equals the product of the numerical values of base § 64. Prop. 17 contains only a special case of 16. After the pro- and altitude, and similarly the number of unit squares contained in blem, Prop. 18. On a yiren straight line to describe a rcclilineal a triangle equals half the product of the munierical values of base figure similar ani similarly situated to a given prelilincal figure, and altitude. there follows another fundamental theorem:

This is often stated by saying that the area of a parallelogram is Prop. 19. Similar triangles are to one another in the duplicate equal to the product of the base and the altitude, meaning by this ratio of their homologous sidrs. In other words, the areas of similar product the product of the numerical values, and not the product as triangles are to one another as the squares on homologous siles. defined above in $ 20. This is generalized in

§ 63. Propositions 24 and 26 relate to parallelograms about Prop. 20. Similar polygons may be dirided into the onme number diagonals, such as are considered in Book I., 43. They areof similar triangles, haring the same ratio lo one another that the Prop. 24. Parallelograms about the dinnelor of any parallclo. polygons hare; and the polygons are to one another in the dunlicate gram are similar to the whole parallelogram and to one another; ratio of their homologous sides.

anıl its converse (Prop. 26), If two similar parallelograms hare n $ 65. Prop. 21. Kectilincal figures which are similar to the same comnion angle, and be similarly situated, thcy are about the same rectilineal figure are also similar to each other, is an iminedinte con. diameter. sequence of the definition of similar figures dg similar figures Between these is inserted a piroblem. may be said to be equal iu “shape” but not in "size," we may state Prop. 25. To describe a rectilineal figure which shall be similar to it also thus:

one giren reclilincar figure, and equal to another giren rcctilineal "Pingures which are equal in aliape to a third are equal in shape figure. to each other.

$ 69. Pror. 27 contains a theorem relating to the theory of Prop. 22. If four elraight lines be proporlionals, the similar maxima and minima. We may state it thus : rectilineal figures similarly clcscribeil on incin shall also be pro- Prop 27 il a parallelogran be virided into tụco by a straight line portionals; and if the similar relilincal figures similarly described cuilling the brese, and if on half the base another parallelogran be on four stroight lins le proporlionıls, those straight lincs shall be constructal similar to one of those parts, then this third pxrallelo. proportionals.

gram is greater than the other part. This is essentially the same as the following:

or far greater interest than this general theorem is a special case If - c:d,

of it, where the parallelograins are changed into rectangles, and then

a:/
ca.d.

where one of the parts into which the parallelogram is divided is § 66. Now follows a proposition which has been much discussed easily recognized to be identical with the following :

made a square ; for then the theorein changes into one which is with regard to Euclid's exact meaning in saying that a ratio is com

THEOREN – Of all rectangles which hare the same perimeter the pounda of two other latios, viz. :

square has the greatest arca. Prop. 23. Parallelograms which are cquiangular lo one another, hare to one another the ralio which is compoundcl of the ralios of their

This may also be stated thus:sides.

Theorem. -Of all reclangles which hare the same arca the squaro

has the lcast perineter. The proof of the proposition makes its meaning clear. In symbols

§ 70. The next three propositions contain problems which may the ratio a :c is compounded of the two ratios a :b and b:c, and if

he sail to be solutions of qudratic equations. The first two are, a:- ':0, 6:6 -b":c", then a :c is compounded of a ':0 and like the last, involved in somewhat obscure language. We tran:

si ribe them as follows:If we consider the ratios as numbers, we may say that the ono

Problem. - To describe on a given base a parallelogram, and to ratio is the proluct of those of which it is compounded, or individe it either internally (Prop. 28) or externally (Prop. 29) from symbols,

A point on the base into two parallelograins, of which the one has b d' 8"

a ծ ?"
and

a given size (is equal in area to a given ligure), whilst the other . 6 i

has a given shape is similar to a given pörallelogram). The theorem in Prop. 23 is the foundation of all mensuration of

Jf we express this again in symbols, calling the given base a, the areas. From it we see at once that two rectangles have the ratio

one part x, and the aliitude y, vre have to deternine x and y in the of their areas compounded of the ratios of their sides.

first case from the equations If A is the area of a rectangle contained by a and I, and B that

(Q - .)y k, of a rectangle contained by c and d, so that Å ab, B cd, then A: B - ab : cd, and this is, the theorem say's, com pounded of the

Y ? ratios a :c and b:d. In forms of quotients,

ko being the given size of the first, and and

P ab

the base and alti.

? co cd

tude of the parallelograin which deterinine the shape of the second

of the required parallelograms. This shows how to multiply quotients in our geometrical calculus. If we substitute the value of y, we get Further, Tico triangles hace lhe ratios of their arcas compounded of the ralios of their berses and thcir allitude. For a triangle is cqnal

9 in area to half a parallelogram which has the same base and the same altitude.

or, To form in which

ax - x - , given, we assume a straight line u as our unit of length (generalls ! where a and 0- are known qnantities, taking 6_pk* an inch, a foot, a mile, &c.), and determine the number a whici expresses how often u is contained in a line a, so that a denotes the The second case (Prop. 29) gives rise, in the same manner, to ratio a : u whether commensurable or not, and that a - ai. We the quadratic call this number a the numerical value of a. If in the same manner

ac + 2*-. B he the numerical value of a line 6 we have

The next problema:b-a:B;

Prop. 30. "To cut a given straight line in extreme and can ratio, in worils : The ratio of luo lincs (and of two like quantilics in

leads to the equation general) is equal to thni of their numcrical rnlus.

az + x? a? This is easily provedl 'hy observing that ro - all, b - Bil, there- This is, therefore, only a special case of the last, and is, besides, fore a :b -- au: Bil, anit this may without difficulty be shown to an old acquaintance, being essentially the same problem as that equal a : 6

propcred in Il. 11.

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