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others: thus, the angle B AC is distin. In the adjoined guished from BAD and DAC. In figure, A is a rightthis case the middle letter, as A, always angled triangle; B an denotes the angular point.
obtuse-angled trian10. (Euc. i. def. 10.) When a gle; and C an acutestraight line standing upon another angled triangle. straight line makes the adjacent angles 16. A triangle is equal to one another, each of them is also said to be equilateral, when its called a right angle, and the straight three sides are equal to one another ; line which stands upon the other is called isosceles, when only two a perpendicular to it.
of its sides are equal; 11. If an angle be
and scalene, when it has not right, it is called
three unequal sides. D oblique. An oblique
is an equilateral triangle, angle is said to be
E an isosceles triangle, acute or obtuse, ac
and Fa scalene triangle. cording as it is less
The three sides of any the same or greater than a right angle.
triangle are frequently distinguished by In the adjoined figure, ABC is a right giving to one of them the name of base, angle, D B C an acute angle, and EBC in which case the other two are called an obtuse angle.
the two sides, and the angular point 12. (Euc. i. def.
opposite to the base is called the vertex 35.) If there be two
or summit. In an isosceles triangle, straight lines in the
considered as such, the vertex is the same plane, which,
angular point between the two equal being produced ever so far both ways, do sides, and the base the side opposite not meet, these straight lines are called to it. parallels.
In a right-angled triangle, the side 13. A plane figure is any portion of a which is opposite to the right angle is plane surface which is included by a line called the hypotenuse; and of the other or lines.
two sides, one is frequently termed the The whole circuit of any figure, that base, and the other the perpendicular. is, the extent of the line or lines by which 17. Of quadrilaterals, a paralleloit is included, is called its perimeter. gram is that which has its opposite sides
14. A plane rectilineal figure is any parallel, as A B C D. A quadrilateral portion of a plane surface, which is in- which has only two of its sides parallel cluded by right lines. These right lines is called a trapezoid, as A BED. are called the sides of the figure, and it A parallelogram, or indeed any quadriis said to be trilateral, or quadrilateral, lateral figure, is sometimes cited by two or multilateral, according as it has three, letters only placed at opposite angles : or four, or a greater number of sides. as “the parallelogram AC”, “the trape
A trilateral figure is more commonly zoid A E.” This plan is never adopted, called a triangle, and a multilateral however, where confusion might ensue figure a polygon.
from it: when used, it must always be It is further to be un
in such a way as to avoid uncertainty ; derstood of rectilineal fi
thus, by “the quadrilateral B D" in the gures in the present trea
adjoined figure, either ABCD or ABED tise, that the several an
might be intended, whereas “the quagles are contained towards the interior of drilateral A C” is distinct from the figure; that is, that they have no such quadrilateral A E." angle as the re-entering angle A in the 18. A rhombus is a parallelogram figure which is adjoined. In other which has two adjoining sides equal.
. words, their perimeters are supposed to be convex externally.
15. A triangle is said to be rightangled, when it has a right angle.
Of triangles which are not right-angled, and which are therefore said to be oblique-angled,—an obtuse-angled triangle is that which has an obtuse angle; and an acute-angled triangle is that 19. A rectangle is a parallelogram which has three acute angles.
which has a right angle. A rectangle
is said to be contained by any two of its 1° the enunciation, declaring what is adjoining sides ; as A Č, which is called to be proved or done ; the rectangle under A B, BC, or the 2°. the construction, inserting the rectangle A B, BC.
lines necessary thereto; 20. A square is a rectangle which has 3°. the demonstration, or course of two adjoining sides equal. The square reasoning ; -And, described upon any straight line AB, 4°. the conclusion, asserting that the or the square of which A B is a side, is thing required has been proved or done. called the square of A B, or A B square, A corollary to any proposition is a
21. The altitude of a parallelogram statement of some truth, which is an obor triangle, is a perpendicular drawn to vious consequence of the proposition. the base from the side or angle opposite. A scholium is a remark or observation.
22. The diagonals of a quadrilateral The object of a problem, as above are the straight lines which join its op- stated, is evidently distinct from that of posite angles.
a theorem. If a problem be regarded, 23. If through a
however, as demonstrating merely the point, E, in the dia
existence of the points and lines required gonal of a paral
in its enunciation, it becomes, for our lelogram, A B C D,
purposes, a theorem certifying the existstraight lines
ence of such. And hence has arisen the drawn parallel to two adjacent sides, introduction of problems into the theory the whole parallelogram will be di- of Geometry; for, the existence of the vided into four quadrilaterals ; of lines and points specified in the con: which two, having the parts of the structions of some theorems not being diagonal for their diagonals, are for that altogether self-evident, it became necesreason said to be about the diagonal; sary, either to introduce distinct problems and the two others, A E, E C, are called for the finding of such, or to point out complements, because, together with the the certainty of their existence by the portions about the diagonal, they com- way of theorem and corollary, as occaplete the whole parallelogram A B C D. sion offered. 24. A circle is a plane
The former plan, exemplified in figure contained by one
Euclid's Elements, has been followed by line, which is called the
the greater number of geometrical circumference, and is
writers; although the problems introsuch that all straight
duced have not, in all cases, been limited lines drawn from a cer
to the very few which are necessary to tain point within the figure to the cir- support the theory. To avoid thus sacumference are equal to one another. crificing unity of purpose, and at the This point is called the centre of the same time not to be wanting to the ends circle; and the distance from the centre of practical geometry, the problems in to the circumference is called the ra- the present treatise have been altogether dius, or, sometimes, the semidiameter, separated from the theorems; and the because it is the half of a straight line requisite support has been supplied to passing through the centre and termi- the latter, in the second of the two ways nated both ways by the circumference, above mentioned. which straight line is called a diameter. The existence of the following lines, &o.
The point C is the centre of the circle will be taken for granted; and they will, A BD; AB is a diameter; and AC a therefore, be referred to by the name of radius or semidiameter.
POSTULATES.* The truths and questions of Geometry
1. A straight line, which joins or are, for the sake of perspicuity, stated and passes through two given points, A, B. considered in small separate discourses called Propositions ; it being proposed in them either to demonstrate something which is asserted, a proposition of which 2. A circle, which is kind is called a theorem, or to show the described from a given manner of doing something which is centre, C, with a given required to be done, a proposition of radius, c'A. which kind is called a problem. A proposition has commonly the fol
* Things required ; from the Latin postulo, to lowing parts :
3. A point which bi
of it; the two together shall be double sects a given finite A
of the third magnitude. straight line, AB,
10. Straight lines which pass through that is, which divides it into two equal the same two points lie in one and the parts.
same straight line.
11. Magnitudes, which may be made to coincide with one another, that is, to
fill exactly the same space, are equal to 4. A straight line
one another. which bisects a given
The converse of this last axiom is like. angle, B AC.
wise true of some magnitudes. In what follows, it will be assumed, with regard
to straight lines and angles; i. e. it will 5. A perpendicular to
he assumed that if two straight lines are a given straight line,
equal, they may be made to coincide erected from
with one another, and the same of two a given point in the same.
B 6. A straight line, which makes with SECTION 2. First Theorems. a given straight line, A B, at a given PROP. 1. (Euc. i. Ax. 11.) point, A, an angle equal to a given rectilineal angle, C.
All right angles are
Let the angles A B C,
the angle D E F. The following truths require no steps
Produce C B to any of reasoning to establish their evidence. point G, and F E to any It may be said of them, that no demon- point H. Then, because stration can make them more evident A B C is a right angle, than they are already, without it: they it is equal to the adjacent angle A BG are, therefore, called self-evident truths (def. 10.); and because DE F is a right or axioms. They will be found of per- angle, it is equal to DE H. petual recurrence in demonstrating the
From E draw any straight line E K. propositions of the following sections, and Then, because the angle KEH is greater are therefore here premised:
than D EH, and that DEH is equal to
DEF, KEH is greater than DEF: AXIOMS.
but D E F is greater than KEF: much 1. Things, which are equal to the more, then, is K E H greater than KE F. same, are equal to one another.
Now, let the angle A B C be applied 2. If equals be added to equals, the to the angle D E F, so that the point B wholes are equal.
upon E, and the straight line BC 3. If equals be taken from equals, upon E F; then (ax. 10.) B G will cointhe remainders are equal.
cide with E H. And, B G coinciding with 4. The doubles of equals are equal.
EH, B A must also coincide with ED; 5. The halves of equals are equal. for, should it fall otherwise, as E K, the
6. The greater of two magnitudes, angle ABG would be greater than increased or diminished by any magni- A B C, by what has been already demon tude, is greater than the less increased strated, whereas, they are equal to one or diminished by the same magnitude.
another. 7. The double of the greater is greater
Therefore, B A coincides with ED, than the double of the less.
and the angle A B C coincides with the 8. The half of the greater is greater angle DEF; and (ax. 11.) is equal to it. than the half of the less.
Therefore, all right angles are equal to 9. If there be two magnitudes, and a one another, which was to be demonthird, and if one of them exceed the strated. third by as much as the other falls short
PROP. 2. (Euc. i. 13 & 14.)
The adjacent angles, which * Authorities, or things having authority; from a
straight line makes with another upon
one side of it, are either two right Cor. 2. Any angle of a triangle is angles, or are together equal to two less than two right angles. right angles : and, conversely, if the adjacent angles, which one straigħt line
PROP. 3. (Euc. i. 15.) makes with two others at the same If two straight lines cut one another, point, be togсther equal to two right the vertical or opposite angles shall be. angles, these two straight lines shall be equal. in one and the same straight line.
Let the two straight Let the straight line
lines A B, CD, cut A B make with CD
one another in the upon one side of it,
point E: the vertical D the adjacent angles
angles A ED, BEC, ABC, ABD: these
as also the vertical are either two right angles, or are to- angles A EC, BE D, shall be equal to gether equal to two right angles. one another.
For, if they are equal, then is each of Because the angles A EC, AED them (def. 10.) a right angle.
are adjacent angles made by the straight But, if not, from the point B draw line À E with C D, they are (2.) toBE perpendicular to CD (Post. 5.). And gether equal to two right angles; and because the angle EBD is equal to the for the like reason, the angles AEC, two angles, EBA, A BD, to each of CE B, are together equal to two right these equals add the angle E B C: angles; therefore, (ax. 1.) the angles therefore, (ax. 2.) the two angles EBC, AEC, A E D together are equal to the EBD are equal to the three angles angles AEC, CE B together. From EBC, EBA, ABD. And in the each of these equals take the angle same manner it may be shewn, that the AEC, and the angle A E D is equal to two angles ABC, ABD, are equal to the angle CE B. (ax. 3.) In the same the same three angles. Therefore, (ax. 1.) manner it may be shown that the angles the angles A B Č, A BD, are together A EC, BE D are equal to one another. equal to the angles E B C, E B D, that is, Therefore, &c. to two right angles.
Cor. (Euc.i. 15. Cor. 2.) If any number Next, let the straight line A B make of straight lines pass through the same with the two straight lines, B C, B D, at point, all the angles about that point, the same point B, the adjacent angles (made by each with that next to it,) shall ABC, A B D together equal to two right be together equal to four right angles. angles : BC, B D shall be in one and
PROP. 4. (Euc. i. 4.) the same straight line. For, let B F be in the same straight
If two triangles have two sides of the line with B C: then, by the first part of one equal to two sides of the other, each the proposition, because AB makes to each, and likewise the included angles angles with C F upon one side of it, these equal ; their other angles shall be equal, angles, viz. A B C, A BF, are together each to each, viz. those to which the equal to two right angles. But ÅBC, equal sides are opposite, and the base, A B D are also equal to two right angles; or third side, of the one shall be equal to therefore, (ax. 1.) A B C, A B D together the base, or third side, of the other. are equal to ABC, ABF together;
Let A B C, DEF be two triangles, and, ABC being taken from each of which have the two_sides A B, A C, these
equals, the angle A B D is equal to equal to the two sides D E, D F, each to ABF (ax. 3.) Therefore B D coincides each, viz. A B to D Е, and AC to DF, with BF; that is, it is in the same
and let them likewise have the angle straight line with B C.
BAC equal to the angle EDF: their Therefore, &c.*
other angles shall be equal, each to Cor. 1. If from a point in a straight each, viz. A B C to DEF, and AC B line there be drawn any number of to D FE, and the straight lines upon one side of it, all the base B C shall be angles (made by each with that next to qual to the base it) shall be together equal to two right EF. angles.
For if the triangle B
ABC be applied to * Hence ihe adjacent angle ABD is sometimes said the triangle D E F, so to be supplementary to ABC; one angle being called that the point A may the supplement of another, when together with that other it is equal to tiro right angles,
be upon D, and the
straight line A B upon D E, the straight equal to them; and the angle B AC line AC will coincide with DF, because coincides with the angle E D F, and is the angle B A C is equal to EDF. Also equal to it (ax. 11.). the point B will coincide with E, because Therefore, &c. AB is equal to D Е, and the point C Cor. The two triangles are equal also with F, because A C is equal to DF; as to surface. and, because the points B, C, coincide
PROP. 6. (Euc. i. 5 & 6.) with the points E, F, the straight line BC coincides with the straight line EF If two sides of a triangle be equal to (ax. 10.), and (ax. 11.) is equal to it; the one another, the opposite angles shall angle A B C coincides with D EF, and be likewise equal : and conversely, if is equal to it; and the angle AC B with two angles of a triangle be equal to one the angle D F E, and is equal to it. another, the opposite sides Therefore, &c.
shall be likewise equal. Cor. The two triangles are equal also
Let A B C be an isosceles as to surface.
triangle, having the side A B Scholium.
equal to the side A C; the It is indifferent which of the two trian- angle A CB shall be equal gles DEF be taken, although in these to the angle A B C. triangles the side D E lie in opposite di- Let the angle BAC be divided into rections from DF; viz. to the right of it in two equal angles by the straight line the one, and to the left of it in the other. AD, which meets the base B C in D
The same may be observed of the next (Post. 4). Then, because the triangles proposition, and of all cases of plane tri- ADB, ADC have two sides of the angles, which are equal in every respect. one equal to two sides of the other, each
to each, and the interjacent angles PROP.5. (Euc. i. 26, first part of.)
BAD, CAD equal to one another, If two triangles have two angles of their other angles are equal, each to the one equal to two angles of the other, each (4.); therefore the angle A C B is each to each, and likewise the interja, equal to A B C. cent* sides equal ; their other sides shall
Next, let the angle A B C be equal be equal, each to each, viz. those to to the angle ACB: the side A C shall which the equal angles are opposite, be equal to the side A B. and the third angle of the one shall be From D, the middle point of BC, equal to the third angle of the other. Let ABC, D E F (see the last figure) and 5.): and, if it do not pass through
erect a perpendicular to BC (Post. 3. be two triangles which have the two the vertex A, let this perpendicular, if angles ABC, ACB of the one, equal possible, cut one of the sides as A B in to the two angles D E F, DFE of the E, and join E C. Then, because the triother, each to each, and likewise the angles É DB, EDC have two sides of side B C equal to the side E F: their the one equal to two sides of the other, other sides shall be equal, each to each, each to each, and the included angles and the third angle BAC shall be EDB, EDC equal to one another equal to the third angle EDF.
(def. 10.), their other angles are equal, For, if the triangle A B C be applied each to each (4.). Therefore the angle to the triangle D E F, so that the point B ECD is equal to EBD. But EBD or may
be upon E, and the straight line BC ABD is equal to ACB: therefore the anupon E F, the point C will coincide with gle ECD is equal to ACB (ax. 1.), the the point F, because B C is equal to EF. less to the greater, which is impossible. Also the straight line B A will coincide Therefore the perpendicular at D cannot in direction with E D, because the angle pass otherwise than through the vertex CBA is equal to FED, and the A: and because the triangles ADB, straight line C A with Fh, because the ADC are equal, according to Prop. 4., angle B C A is equal to EFD. But, if the side A B is equal to the side A C. two straight lines which cut one an
Therefore, &c. other, coincide with other two which
Cor. 1. Every equilateral triangle is cut one another, it is manifest that the also equiangular; and conversely. points of intersection must likewise coin
Cor. 2. In an isosceles triangle ABC, cide. Therefore, the point A coincides if the equal sides A B, A C, be prowith D, and the sides A B, A C, coin- duced, the angles upon the other side of cide with the sides DE, DF, and are the base B C will be equal to one an.
Interjacent sides," i. e. sides lying between. other; for, each of them together with