# Book I.

## Definitions.

### I.

A point is that which has no parts.

### II.

A line is length without breadth.

### III.

The extremities of a line are points.

### IV.

A ſtraight or right line is that which lies evenly between its extremities.

### V.

A ſurface is that which has length and breadth only.

### VI.

The extremities of a ſurface are lines.

### VII.

A plane ſurface is that which lies evenly between its extremities.

### VIII.

A plane angle is the inclination of two lines to one another, in a plane, which meet together, but are not in the ſame direction.

### IX.

A plane rectilinear angle is the inclination of two ſtraight lines to one another, which meet together, but are not in the ſame ſtraight line.

### X.

When one ſtraight line ſtanding on another ſtraight line makes the adjacent angles equal, each of theſe angles is called a right angle, and each of theſe lines is ſaid to be perpendicular to the other.

### XI.

An obtuſe angle is an angle greater than a right angle.

### XII.

An acute angle is leſs than a right angle.

### XIII.

A term or boundary is the extremity of any thing.

### XIV.

A figure is a ſurface encloſed on all ſides by a line or lines.

### XV.

A circle is a plane figure, bounded by one continued line, called its circumference or periphery; and having a certain point within it, from which all ſtraight lines drawn to its circumference are equal.

### XVI.

This point (from which the equal lines are drawn) is called the centre of the circle.

### XVII.

A diameter of a circle is a ſtraight line drawn through the centre, terminated both ways in the circumference.

### XVIII.

A ſemicircle is the figure contained by the diameter, and the part of the circle cut off by the diameter.

### XIX.

A ſegment of a circle is a figure contained by a ſtraight line, and the part of the circumference which it cuts off.

### XX.

A figure contained by ſtraight lines only, is called a rectilinear figure.

### XXI.

A triangle is a rectilinear figure included by three ſides.

### XXII.

A quadrilateral figure is one which is bounded by four ſides. The ſtraight lines and connecting the vertices of the oppoſite angles of a quadrilateral figure, are called its diagonal.

### XXIII.

A polygon is a rectilinear figure bounded by more than four ſides.

### XXIV.

A triangle whoſe three ſides are equal, is ſaid to be equilateral.

### XXV.

A triangle which has only two ſides equal is called an iſoſceles triangle.

### XXVI.

A ſcalene triangle is one which has no two ſides equal.

### XXVII.

A right angled triangle is that which has a right angle.

### XXVIII.

An obtuſe angled triangle is that which has an obtuſe angle.

### XXIX.

An acute angled triangle is that which has three acute angles.

### XXX.

Of four-ſided figures, a ſquare is that which has all its ſides equal, and all its angles right angles.

### XXXI.

A rhombus is that which has all its ſides equal, but its angles are not right angles.

### XXXII.

An oblong is that which has all its angles right angles, but has not all its ſides equal.

### XXXIII.

A rhomboid is that which has its oppoſite ſides equal to one another, but all its ſides are not equal, nor its angles right angles.

### XXXIV.

All other quadrilateral figures are called trapeziums.

### XXXV.

Parallel ſtraight lines are ſuch as are in the ſame plane, and which being produced continually in both directions, would never meet.

## Postulates.

### I.

Let it be granted that a ſtraight line may be drawn from any one point to any other point.

### II.

Let it be granted that a finite ſtraight line may be produced to any length in a ſtraight line.

### III.

Let it be granted that a circle may be deſcribed with any centre at any diſtance from that centre.

## Axioms.

### I.

Magnitudes which are equal to the ſame are equal to each other.

### II.

If equals be added to equals the ſums will be equal.

### III.

If equals be taken away from equals the remainders will be equal.

### IV.

If equals be added to unequals the ſums will be unequal.

### V.

If equals be taken away from unequals the remainders will be unequal.

### VI.

The doubles of the ſame or equal magnitudes are equal.

### VII.

The halves of the ſame or equal magnitudes are equal.

### VIII.

Magnitudes which coincide with one another, or exactly fill the ſame ſpace, are equal.

### IX.

The whole is greater than its part.

### X.

Two ſtraight lines cannot include a ſpace.

### XI.

All right angles are equal.

### XII.

If two ſtraight lines ( ) meet a third ſtraight line () ſo as to make the two interior angles ( and ) on the ſame ſide leſs than two right angles, theſe two ſtraight lines will meet if they be produced on that ſide on which the angles are leſs than two right angles.

The twelfth axiom may be expreſſed in any of the following ways:

1. Two diverging ſtraight lines cannot be both parallel to the ſame ſtraight line.
2. If a ſtraight line interſect one of the two parallel ſtraight lines it muſt also interſect the other.
3. Only one ſtraight line can be drawn through a given point, parallel to a given ſtraight line.

## Elucidations.

Geometry has for its principal objects the expoſition and explanation of the properties of figure, and figure is defined to be the relation which ſubſiſts between the boundaries of ſpace. Space or magnitude is of three kinds, linear, ſuperficial, and ſolid.

Angles might properly be conſidered as a fourth ſpecies of magnitude. Angular magnitude evidently conſiſts of parts, and muſt therefore be admitted to be a ſpecies of quantity. The ſtudent muſt not ſuppoſe that the magnitude of an angle is affected by the length of the ſtraight lines which include it, and of whoſe mutual divergence it is the meaſure. The vertex of an angle is the point where the ſides or the legs of the angle meet, as A.

An angle is often deſignated by a ſingle letter when its legs are the only lines which meet together at its vertex. Thus the red and blue lines form the yellow angle, which in other ſyſtems would be called the angle A. But when more than two lines meet in the ſame point, it was neceſſary by former methods, in order to avoid confuſion, to employ three letters to deſignate an angle about that point, the letter which marked the vertex of the angle being always placed in the middle. Thus the black and red lines meeting together at C, form the blue angle, and has been uſually denominated the angle FCD or DCF. The lines FC and CD are the legs of the angle; the point C is its vertex. In like manner the black angle would be deſignated the angle DCB or BCD. The red and blue angles added together, or the angle HCF added to FCD, make the angle HCD; and ſo of the other angles.

When the legs of an angle are produced or prolonged beyond its vertex, the angles made by them on both ſides of the vertex are ſaid to be vertically oppoſite to each other: Thus the red and yellow angles are ſaid to be vertically oppoſite angles.

Superpoſition is the proceſs by which one magnitude may be conceived to be placed upon another, ſo as exactly to cover it, or ſo that every part of each ſhall exactly coincide.

A line is ſaid to be produced, when it is extended, prolonged, or has its length increaſed, and the increaſe of length which it receives is called its produced part, or its production.

The entire length of the line or lines which encloſe a figure, is called its perimeter. The firſt ſix books of Euclid treat of plane figures only. A line drawn from the centre of a circle to its circumference, is called a radius. The lines which include a figure are called its ſides. That ſide of a right angled triangle, which is oppoſite to the right angle, is called the hypotenuſe. An oblong is defined in the ſecond book, called a rectangle. All the lines which are conſidered in the firſt ſix books of the Elements are ſuppoſed to be in the ſame plane.

The ſtraight-edge and compaſſes are the only inſtruments, the uſe of which is permitted in Euclid, or plane Geometry. To declare this reſtriction is the object of the poſtulates.

The Axioms of geometry are certain general propoſitions, the truth of which is taken to be ſelf-evident and incapable of being eſtablished by demonſtration.

Propoſitions are thoſe reſults which are obtained in geometry by a proceſs of reaſoning. There are two ſpecies of propoſitions in geometry, problems and theorems.

A Problem is a propoſition in which ſomething is propoſed to be done; as a line to be drawn under ſome given conditions, a circle to be deſcribed, ſome figure to be conſtructed, &c.

The ſolution of the problem conſiſts in ſhowing how the thing required may be done by the aid of the rule or ſtraight-edge and compaſſes.

The demonſtration conſiſts in proving that the proceſs indicated in the ſolution really attains the required end.

A Theorem is a propoſition in which the truth of ſome principle is aſſerted. This principle muſt be deduced from the axioms and definitions, or other truths previously and independently eſtablished. To ſhow this is the object of demonſtration.

A Problem is analogous to a poſtulate.

A Theorem reſembles an axiom.

A Poſtulate is a problem, the ſolution of which is aſſumed.

An Axiom is a theorem, the truth of which is granted without demonſtration.

A Corollary is an inference deduced immediately from a propoſition.

A Scholium is a note or obſervation on a propoſition not containing an inference of ſufficient importance to entitle it to the name of a corollary.

A Lemma is a propoſition merely introduced for the purpoſe of eſtablishing ſome more important propoſition.

## Proposition I. Problem.

On a given finite ſtraight line () to deſcribe an equilateral triangle.

Deſcribe and (poſtulate 3.); draw and (poſt. 1.). then will be equilateral.

For = (def. 15.); and = (def. 15.), = (axiom. 1.);

and therefore is the equilateral triangle required.

Q. E. D.

## Proposition II. Problem.

From a given point ( ), to draw a ſtraight line equal to a given finite ſtraight line ()

Draw (poſt. 1.), deſcribe (pr. 1.), produce (poſt. 2.), deſcribe (poſt. 3.), and (poſt. 3.); produce (poſt. 2.), then is the line required.

For = (def. 15.), and = (conſt.), = (ax. 3.), but (def. 15.) = = ; drawn from the given point ( ), is equal the given line .

Q. E. D.

## Proposition III. Problem.

From the greater ( ) of two given ſtraight lines, to cut off a part equal to the leſs ().

Draw = (pr. 2.); deſcribe (poſt. 3.), then = .

For = (def. 15.), = (conſt.); = (ax. 1.).

Q. E. D.

## Proposition IV. Theorem.

If two triangles have two ſides of the one reſpectively equal to two ſides of the other, ( to and to ) and the angles ( and ) contained by thoſe equal ſides alſo equal; then their baſes or their ſides ( and ) are alſo equal: and the remaining and their remaining angles oppoſite to equal ſides are reſpectively equal ( = and = ): and the triangles are equal in every reſpect.

Let the two triangles be conceived, to be ſo placed, that the vertex of one of the equal angles, or ; ſhall fall upon that of the other, and to coincide with , then will coincide with if applied: conſequently will coincide with , or two ſtraight lines will encloſe a ſpace, which is impoſſible (ax. 10), therefore = , = and = , and as the triangles and coincide, when applied, they are equal in every reſpect.

Q. E. D.

## Proposition V. Theorem.

In any iſoſceles triangle if the equal ſides be produced, the external angles at the baſe are equal, and the internal angles at the baſe are alſo equal.

Produce , and , (poſt. 2.), take = , (pr. 3.); draw and .

Then in and we have,
= (conſt.), common to
both, and = (hyp.) = ,
= and = (pr. 4.).

Again in and we have = ,
= and = ,
= and = (pr. 4.) but
= , ∴ = (ax. 3.)

Q. E. D.

## Proposition VI. Theorem.

In any triangle ( ) if two angles ( and ) are equal, the ſides ( and ) oppoſite two them are alſo equal.

For if the ſides be not equal, let one of them be greater than the other , and from it cut off = (pr. 3.), draw .

Then and , = , (conſt.) = (hyp.) and common, the triangles are equal (pr. 4.) a part equal to the whole, which is abſurd; neither of the ſides or is greater than the other, hence they are equal.

Q. E. D.

## Proposition VII. Theorem.

On the ſame baſe () and on the ſame ſide of it there cannot be two triangles having their conterminous ſides ( and , and ) at both extremities of the baſe, equal to each other.

When two triangles ſtand on the ſame baſe, and on the ſame ſide of it, the vertex of one ſhall either fall outſide of the other triangle, or within it; or, laſtly, on one of its ſides.

If it be poſſible let the two triangles be conſtructed ſo that { = = } , then draw and,

= (pr. 5.) < and < but (pr. 5.) = } which is abſurd,

therefore the two triangles cannot have their conterminous ſides equal at both extremities of the baſe.

Q. E. D.

## Proposition VIII. Theorem.

If two triangles have two ſides of the one reſpectively equal to two ſides of the other ( = and = ), and alſo their baſes ( = ), equal; then the angles ( and ) contained by their equal ſides are alſo equal.

If the equal baſes and be conceived to be placed upon the other, ſo that the triangles ſhall lie at the ſame ſide of them, and that the equal ſides and , and be conterminous, the vertex of the one muſt fall on the vertex of the other; for to ſuppoſe them not coincident would contradict the laſt propoſition.

Therefore the ſides and , being coincident with and ,
= .

Q. E. D.

## Proposition IX. Problem.

To biſect a given rectilinear angle ( ).

Take = (pr. 3.) draw , upon which deſcribe (pr. 1.), draw .

Becauſe = (conſt.) and common to the two triangles and = (conſt.),
= (pr. 8.)

Q. E. D.

## Proposition X. Problem.

To biſect a given finite ſtraight line ( ).

Conſtruct (pr. 1.),
draw , making = (pr. 9.).

Then = by (pr. 4.),
= (conſt.) = and
common to the two triangles.

Therefore the given line is biſected.

Q. E. D.

## Proposition XI. Problem.

From a given point ( ), in a given ſtraight line ( ), to draw a perpendicular.

Take any point ( ) in the given line,
cut off = (pr. 3.),
conſtruct (pr. 1.),
draw and it ſhall be perpendicular to the given line.

For = (conſt.)
= (conſt.)
and common to the two triangles.

Therefore = (pr. 8.)
(def. 10.).

Q. E. D.

## Proposition XII. Problem.

To draw a ſtraight line perpendicular to a given indefinite ſtraight line ( ) from a given (point ) without.

With the given point as centre, at one ſide of the lines, and any diſtance capable of extending to the other ſide deſcribe ,

Make = (pr. 10.)
draw , and .
then .

For (pr. 8.) ſince = (conſt.)
common to both,
and = (def. 15.)

= , and
(def. 10.).

Q. E. D.

## Proposition XIII. Theorem.

When a ſtraight line () ſtanding upon another ſtraight line () makes angles with it; they are either two right angles or together equal to two right angles.

If be to then,
and = (def. 10.),

But if be not to ,
draw ; (pr. 11.)
+ = (conſt.),
= = +
+ = + + (ax. 2.)
= + = .

Q. E. D.

## Proposition XIV. Theorem.

If two ſtraight lines ( and ), meeting a third ſtraight line (), at the ſame point, and at oppoſite ſides of it, make with it adjacent angles ( and ) equal to two right angles; theſe ſtraight lines lie in one continuous ſtraight line.

For, if poſſible, let , and not ,
be the continuation of ,
then + =
but by the hypotheſis + =
= , (ax. 3.); which is abſurd (ax. 9.).
, is not the continuation of , and the like may be demonſtrated of any other ſtraight line except , is the continuation of .

Q. E. D.

## Proposition XV. Theorem.

If two ſtraight lines ( and ) interſect one another, the vertical angles and , and are equal.

+ =
+ =
= .

In the ſame manner it may be ſhown that
=

Q. E. D.

## Proposition XVI. Theorem.

If a ſide of a triangle ( ) is produced, the external angle ( ) is greater than either of the internal remote angles ( or ).

Make = (pr. 10.).
Draw and produce it until
= ; draw .

In and ; =
= and = (conſt. pr. 15.),
= (pr. 4.),
> .

In like manner it can be ſhown, that if
be produced, > , and therefore
which is = is > .

Q. E. D.

## Proposition XVII. Theorem.

Any of two angles of a triangle are together leſs than two right angles.

Produce , then will
+ =

But > (pr. 16.)
+ < ,

and in the ſame manner it may be ſhown that any other two angles of the triangle taken together are leſs than two right angles.

Q. E. D.

## Proposition XVIII. Theorem.

In any triangle if one ſide be greater than another , the angle oppoſite to the greater ſide is greater than the angle to the oppoſite to the leſs. i. e. > .

Make = (pr.3.), draw .

Then will = (pr. 5.);
but > (pr. 16.);
> and much more
is > .

Q. E. D.

## Proposition XIX. Theorem.

If in any triangle one angle be greater than another the ſide which is oppoſite to the greater angle, is greater than the ſide oppoſite the leſs.

If be not greater than then muſt
= or < .

If = then
= (pr. 5.);
which is contrary to the hypotheſis.
is not leſs than ; for if it were,
< (pr. 18.)
which is contrary to the hypotheſis:
> .

Q. E. D.

## Proposition XX. Theorem.

Any two ſides and of a triangle taken together are greater than the third ſide ().

Produce , and
make = (pr. 3.);
draw .

Then becauſe = (conſt.),

= (pr. 5.) > (ax. 9.)

+ > (pr. 19.)
and + > .

Q. E. D.

## Proposition XXI. Theorem.

If from any point ( ) within a triangle ſtraight lines be drawn to the extremities of one ſide (), theſe lines muſt be together leſs than the other two ſides, but muſt contain a greater angle.

Produce ,
+ > (pr. 20.),
+ > + (ax. 4.)

In the ſame manner it may be ſhown that
+ > + ,
+ > + ,
which was to be proved.

Again > (pr. 16.), and also > (pr. 16.), > .

Q. E. D.

## Proposition XXII. Theorem.

Given three right lines { the ſum of any two greater than the third, to conſtruct a triangle whoſe ſides ſhall be reſpectively equal to the given lines.

Aſſume = (pr. 3.). Draw = and = } (pr. 2.).

describe and (poſt. 3.);
draw and ,
then will be the triangle required.

For = , = = , and = = . } (conſt.)

Q. E. D.

## Proposition XXIII. Problem.

At a given point ( ) in a given ſtraight line ( ), to make an angle equal to a given rectilineal angle ( ).

Draw between any two points in the legs of the given angle.

Conſtruct (pr. 22.). ſo that
= , =
and = .

Then = (pr. 8.).

Q. E. D.

## Proposition XXIV. Theorem.

If two triangles have two ſides of one reſspectively equal to two ſides of the other ( to and to ), and if one of the angles ( ) contained by the equal ſides be greater than the other ( ), the ſide () which is oppoſite to the greater angle is greater than the ſide () which is oppoſite to the leſs angle.

Make = (pr. 23.),
and = (pr. 3.),
draw and .
Becauſe = (ax. 1. hyp. conſt.)
= (pr. 5.)
but <
and < ,
> (pr. 19.)
but = (pr. 4.)
> .

Q. E. D.

## Proposition XXV. Theorem.

If two triangles have two ſides ( and ) of the one reſpectively equal to two ſides ( and ) of the other, but their baſes unequal, the angle ſubtended by the greater baſe () of the one, muſt be greater than the angle ſubtended by the leſs baſe () of the other.

=, > or < is not equal to
for if = then = (pr. 4.)
which is contrary to the hypotheſis;

is not leſs than
for if <
then < (pr. 24.),
which is alſo contrary to the hypotheſis:

> .

Q. E. D.

## Proposition XXVI. Theorem.

If two triangles have two angles of the one reſpectively equal to two angles of the other, ( = and = ), and a ſide of the one equal to a ſide of the other ſimilarly placed with reſpect to the equal angles, the remaining ſides and angles are reſpectively equal to one another.

### Case I.

Let and which lie between the equal angles be equal,
then = .
For if it be poſſible, let one of them be greater than the other;
make = , draw .

In and we have
= , = , = ;
= (pr. 4.)
but = (hyp.)
and therefore = , which is abſurd; hence neither of the ſides and is greater than the other; and ∴ they are equal;
= , and = , (pr. 4.).

### Case II.

Again, let = , which lie oppoſite the equal angles and . If it be poſſible, let > , then take = , draw .

Then in and we have = ,
= and = ,
= (pr. 4.)
but = (hyp.)
= which is abſurd (pr. 16.).

Conſequently, neither of the ſides or is greater than the other, hence they muſt be equal. It follows (by pr. 4.) that the triangles are equal in all reſpects.

Q. E. D.

## Proposition XXVII. Theorem.

If a ſtraight line () meeting two other ſtraight line, ( and ) makes with them the alternate angles and ; and ) equal, theſe two ſtraight lines are parallel.

If be not parallel to they ſhall meet when produced.

If it be poſſible, let thoſe lines be not parallel, but meet when produce; then the external angle is greater than (pr. 16), but they are alſo equal (hyp.), which is abſurd: in the ſame manner it may be ſhown that they cannot meet on the other ſide; ∴ they are parallel.

Q. E. D.

## Proposition XXVIII. Theorem.

If a ſtraight line (), cutting two other ſtraight lines ( and ), makes the external equal to the internal and oppoſite angle, at the ſame ſide of the cutting line (namely, = or = ), or if it makes the two internal angles at the ſame ſide ( and , or and ) together equal to two right angles, thoſe two ſtraight lines are parallel.

Firſt, if = , then = (pr. 15.),
= (pr. 27.).

Secondly, if + = ,
then + = (pr. 13.),
+ = + (ax. 3.)
=
(pr. 27.)

Q. E. D.

## Proposition XXIX. Theorem.

A ſtraight line () falling on two parallel ſtraight lines ( and ), makes the alternate angles equal to one another; and alſo the external equal to the internal and oppoſite angle on the ſame ſide; and the two internal angles on the ſame ſide together equal to two right angles.

For if the alternate angles and be not equal, draw , making = (pr. 23).

Therefore (pr. 27.) and therefore two ſtraight lines which interſect are parallel to the ſame ſtraight line, which is impoſſible (ax. 12).

Hence the alternate angles and are not unequal, that is, they are equal: = (pr. 15); = , the external angle equal to the internal and oppoſite on the ſame ſide: if be added to both, then + = = (pr. 13). That is to ſay, the two internal angles at the ſame ſide of the cutting line are equal to two right angles.

Q. E. D.

## Proposition XXX. Theorem.

Straight lines ( ) which are parallel to the ſame ſtraight line (), are parallel to one another.

Let interſect { } ;
Then, = = (pr. 29.),
=
(pr. 27.)

Q. E. D.

## Proposition XXXI. Problem.

From a given point to draw a ſtraight line parallel to a given ſtraight line ().

Draw from the point to any point in ,
make = (pr. 23.),
then (pr. 27.).

Q. E. D.

## Proposition XXXII. Problem.

If any ſide () of a triangle be produced, the external angle ( ) is equal to the ſum of the two internal and oppoſite angles ( and ), and the three internal angles of every triangle taken together are equal to two right angles.

Through the point draw
(pr. 31.).

Then { = = } (pr. 29.),

+ = (ax. 2.),
and therefore
+ + = = (pr. 13.).

Q. E. D.

## Proposition XXXIII. Theorem.

Straight lines ( and ) which join the adjacent extremities of two equal and parallel ſtraight lines ( and ), are themſelves equal and parallel.

Draw the diagonal.
= (hyp.)
= (pr. 29.)
and common to the two triangles;
= , and = (pr. 4.);
and (pr. 27.).

Q. E. D.

## Proposition XXXIV. Theorem.

The oppoſite ſides and angles of any parallelogram are equal, and the diagonal () divides it into two equal parts.

Since { = = } (pr. 29.) and common to the two triangles.

{ = = = } (pr. 26.)
and = (ax.):

Therefore the oppoſite ſides and angles of the parallelogram are equal: and as the triangles and are equal in every reſpect (pr. 4,), the diagonal divides the parallelogram into two equal parts.

Q. E. D.

## Proposition XXXV. Theorem.

Parallelograms on the ſame baſe, and between the ſame parallels, are (in area) equal.

On account of the parallels, = ; = ; = } (pr. 29.) (pr. 29.) (pr. 34.)

But, = (pr. 8.)
minus = ,
and minus = ;
= .

Q. E. D.

## Proposition XXXVI. Theorem.

Parallelograms ( and ) on equal baſes, and between the ſame parallels, are equal.

Draw and ,
= = , by (pr. 34, and hyp.);
= and ;
= and (pr. 33.)

And therefore is a parallelogram:
but = = (pr. 35.)
= (ax. 1.).

Q. E. D.

## Proposition XXXVII. Theorem.

Triangles and on the ſame baſe () and between the ſame parallels are equal.

Draw } (pr. 31.)

Produce .

and are parallelograms on the ſame baſe, and between the ſame parallels, and therefore equal. (pr. 35.)

{ = twice = twice } (pr. 34.)

= .

Q. E. D.

## Proposition XXXVIII. Theorem.

Triangles ( and ) on equal baſes and between the ſame parallels are equal.

Draw } (pr. 31.)

= (pr. 36.);
= twice (pr. 34.),
and = twice (pr. 34.),
= (ax. 7.).

Q. E. D.

## Proposition XXXIX. Theorem.

Equal triangles and on the ſame baſe () and on the ſame ſide of it, are between the ſame parallels.

If , which joins the vertices of the triangles, be not , draw (pr. 31.), meeting .

Draw .

Becauſe (conſt.)
= (pr. 37.):
but = (hyp.);

= , a part equal to the whole, which is abſurd.

; and in the ſame manner it can be demonſtrated, that no other line except
is ; .

Q. E. D.

## Proposition XL. Theorem.

Equal triangles ( and ) on equal baſes, and on the ſame ſide, are between the ſame parallels.

If which joins the vertices of triangles
be not ,
draw (pr. 31.),
meeting .

Draw .
Becauſe (conſt.)
= but =
= , a part equal to the whole, which is abſurd.
: and in the ſame manner it can be demonſtrated, that no other line except
is : .

Q. E. D.

## Proposition XLI. Theorem.

If a parallelogram and a triangle are upon the ſame baſe and between the ſame parallels and , the parallelogram is double the triangle.

Draw the diagonal;
Then = (pr. 37.)
= twice (pr. 34.)
= twice .

Q. E. D.

## Proposition XLII. Theorem.

To conſtruct a parallelogram equal to a given triangle and having an angle equal to a given rectilinear angle .

Make = (pr. 10.)
Draw .
Make = (pr. 23.)

Draw { } (pr. 31.)

= twice (pr. 41.)
but = (pr. 38.)
= .

Q. E. D.

## Proposition XLIII. Theorem.

The complements and of the parallelograms which are about the diagonal of a parallelogram are equal.

= (pr. 34.)
and = (pr. 34.)
= (ax. 3.)

Q. E. D.

## Proposition XLIV. Problem.

To a given ſtraight line () to apply a parallelogram equal to a given triangle ( ), and having an angle equal to a given rectilinear angle ( ).

Make = with = (pr. 42.)
and having one of its ſides conterminous with and in continuation of . Produce till it meets draw produce it till it meets continued; draw meeting produced, and produce .

= (pr. 43.)
but = (conſt.)
= ; and
= = = (pr. 29. and conſt.)

Q. E. D.

## Proposition XLV. Problem.

To conſtruct a parallelogram equal to a given rectilinear figure ( ) and having an angle equal to a given rectilinear angle ( ).

Draw and dividing the rectilinear figure into triangles.

Conſtruct =
having = (pr. 42.)
to apply =
having = (pr. 44.)
to apply =
having = (pr. 44.)
=
and is a parallelogram (prs. 29, 14, 30.)
having = .

Q. E. D.

## Proposition XLVI. Problem.

Upon a given ſtraight line () to conſtruct a ſquare.

Draw and = (pr. 11. and 3.)

Draw , and meeting drawn .

In = (conſt.)
= a right angle (conſt.)
= = a right angle (pr. 29.),
and the remaining ſides and angles muſt be equal. (pr. 34.)
and is a ſquare. (def. 27.)

Q. E. D.

## Proposition XLVII. Theorem.

In a right angled triangle the ſquare on the hypotenuſe is equal to the ſum of the ſquares of the ſides, ( and ).

On , and deſcribe ſquares, (pr. 46.)

Draw (pr. 31.) alſo draw and .

= ,

= and = ;

= .

Again, becauſe
= twice ,
and = twice ;
= .

In the ſame manner it may be ſhown
that = ;
hence = .

Q. E. D.

## Proposition XLVIII. Theorem.

If the ſquare of one ſide () of a triangle is equal to the ſquares of the other two ſides ( and ), the angle ( ) ſubtended by that ſide is a right angle.

Draw and = (prs. 11. 3.)
and draw alſo.

Since = (conſt.)
2 = 2;
2 + 2 = 2 + 2,
but 2 + 2 = 2 (pr. 47.),
and 2 + 2 = 2 (hyp.)
2 = 2,
= ;
and = (pr. 8.),
conſequently is a right angle.

Q. E. D.