« AnteriorContinuar »
others: thus, the angle B A C is distinguished from BAD and DAC. In this case the middle letter, as A, always denotes the angular point.
10. (Euc. i. def. 10.) When a straight line standing upon another straight line makes the adjacent angles equal to one another, each of them is called a right angle, and the straight line which stands upon the other is called a perpendicular to it.
11. If an angle be not right, it is called oblique. An oblique angle is said to be acute or obtuse, according as it is less
or greater than a right angle. In the adjoined figure, ABC is a right angle, D B C an acute angle, and E BC an obtuse angle.
12. (Euc. i. def. 35.) If there be two straight lines in the same plane, which,
being produced ever so far both ways, do not meet, these straight lines are called parallels.
13. A plane figure is any portion of a plane surface which is included by a line or lines.
The whole circuit of any figure, that is, the extent of the line or lines by which it is included, is called its perimeter.
14. A plane rectilineal figure is any portion of a plane surface, which is included by right lines. These right lines are called the sides of the figure, and it is said to be trilateral, or quadrilateral, or multilateral, according as it has three, or four, or a greater number of sides.
A trilateral figure is more commonly called a triangle, and a multilateral figure a polygon.
It is further to be understood of rectilineal figures in the present treatise, that the several an
gles are contained towards the interior of the figure; that is, that they have no such angle as the re-entering angle A in the figure which is adjoined. In other 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 which has three acute angles.
In a right-angled triangle, the side which is opposite to the right angle is called the hypotenuse; and of the other two sides, one is frequently termed the base, and the other the perpendicular.
17. Of quadrilaterals, a parallelogram is that which has its opposite sides parallel, as ABCD. A quadrilateral which has only two of its sides parallel is called a trapezoid, as A B ED.
A parallelogram, or indeed any quadrilateral figure, is sometimes cited by two letters only placed at opposite angles: as "the parallelogram AC", "the trapezoid A E." This plan is never adopted, however, where confusion might ensue from it when used, it must always be in such a way as to avoid uncertainty; thus, by "the quadrilateral B D" in the adjoined figure, either ABCD or ABED might be intended, whereas "the quadrilateral A C" is distinct from quadrilateral A E."
18. A rhombus is a parallelogram which has two adjoining sides equal.
is said to be contained by any two of its adjoining sides; as A C, which is called the rectangle under A B, BC, or the rectangle AB, B C.
20. A square is a rectangle which has two adjoining sides equal. The square described upon any straight line AB, or the square of which A B is a side, is called the square of A B, or A B square. 21. The altitude of a parallelogram or triangle, is a perpendicular drawn to the base from the side or angle opposite. 22. The diagonals of a quadrilateral are the straight lines which join its opposite angles.
23. If through a point, E, in the diagonal of a parallelogram, A B C D, straight lines be
drawn parallel to two adjacent sides, the whole parallelogram will be divided into four quadrilaterals; of which two, having the parts of the diagonal for their diagonals, are for that reason said to be about the diagonal; and the two others, A E, E C, are called complements, because, together with the portions about the diagonal, they_complete the whole parallelogram ABC D. 24. A circle is a plane figure contained by one line, which is called the circumference, and such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another. This point is called the centre of the circle; and the distance from the centre to the circumference is called the radius, or, sometimes, the semidiameter, because it is the half of a straight line passing through the centre and terminated both ways by the circumference, which straight line is called a diameter. The point C is the centre of the circle ABD; AB is a diameter; and AC a radius or semidiameter.
The truths and questions of Geometry are, for the sake of perspicuity, stated and considered in small separate discourses called Propositions; it being proposed in them either to demonstrate something which is asserted, a proposition of which kind is called a theorem, or to show the manner of doing something which is required to be done, a proposition of which kind is called a problem.
A proposition has commonly the following parts:
1o. the enunciation, declaring what is to be proved or done;
20. the construction, inserting the lines necessary thereto;
30. the demonstration, or course of reasoning; -And,
4°. the conclusion, asserting that the thing required has been proved or done. A corollary to any proposition is a statement of some truth, which is an obvious consequence of the proposition.
A scholium is a remark or observation. The object of a problem, as above stated, is evidently distinct from that of a theorem. If a problem be regarded, however, as demonstrating merely the existence of the points and lines required in its enunciation, it becomes, for our purposes, a theorem certifying the existence of such. And hence has arisen the introduction of problems into the theory of Geometry; for, the existence of the lines and points specified in the constructions of some theorems not being altogether self-evident, it became necessary, either to introduce distinct problems for the finding of such, or to point out the certainty of their existence by the way of theorem and corollary, as occasion offered.
The former plan, exemplified in Euclid's Elements, has been followed by the greater number of geometrical writers; although the problems introduced have not, in all cases, been limited to the very few which are necessary to support the theory. To avoid thus sacrificing unity of purpose, and at the same time not to be wanting to the ends of practical geometry, the problems in the present treatise have been altogether separated from the theorems; and the requisite support has been supplied to the latter, in the second of the two ways above mentioned.
The existence of the following lines, &o. will be taken for granted; and they will, therefore, be referred to by the name of POSTULATES.*
1. A straight line, which joins or passes through two given points, A, B.
3. A point which bi
sects a given finite straight line, A B,
of it; the two together shall be double of the third magnitude.
10. Straight lines which pass through
that is, which divides it into two equal the same two points lie in one and the
6. A straight line, which makes with a given straight line, A B, at a given point, A, an angle equal to a given rectilineal angle, C.
The following truths require no steps of reasoning to establish their evidence. It may be said of them, that no demonstration can make them more evident
than they are already, without it: they are, therefore, called self-evident truths or axioms. They will be found of perpetual recurrence in demonstrating the propositions of the following sections, and are therefore here premised:
1. Things, which are equal to the same, are equal to one another.
2. If equals be added to equals, the wholes are equal.
3. If equals be taken from equals, the remainders are equal.
4. The doubles of equals are equal. 5. The halves of equals are equal. 6. The greater of two magnitudes, increased or diminished by any magnitude, is greater than the less increased or diminished by the same magnitude.
7. The double of the greater is greater than the double of the less.
8. The half of the greater is greater than the half of the less.
9. If there be two magnitudes, and a third, and if one of them exceed the third by as much as the other falls short
* Authorities, or things having authority; from a Greek word.
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 one another.
The converse of this last axiom is likewise true of some magnitudes. In what follows, it will be assumed, with regard to straight lines and angles; i. e. it will be assumed that if two straight lines are equal, they may be made to coincide with one another, and the same of two angles.
SECTION 2. First Theorems.
PROP. 1. (EUc. i. Ax. 11.) All right angles are equal to one another.
Let the angles A B C, DEF be each of them a right angle; the angle ABC shall be equal to the angle D E F.
Produce C B to any point G, and FE to any point H. Then, because ABC is a right angle, it is equal to the adjacent angle A B G (def. 10.); and because DE F is a right angle, it is equal to DE H.
From E draw any straight line E K. Then, because the angle KEH is greater than DEH, and that DE H is equal to DEF, KEH is greater than DEF: but D E F is greater than K EF: much more, then, is K E H greater than KE F.
Now, let the_angle ABC be applied to the angle D E F, so that the point B may be upon E, and the straight line B C upon EF; then (ax. 10.) B G will coincide with E H. And, B G coinciding with EH, BA must also coincide with ED; for, should it fall otherwise, as EK, the angle ABG would be greater than AB C, by what has been already demon strated, whereas, they are equal to one
one side of it, are either two right angles, or are together equal to two right angles: and, conversely, if the adjacent angles, which one straight line makes with two others at the same point, be together equal to two right angles, these two straight lines shall be in one and the same straight line.
Let the straight line AB make with CD upon one side of it,
the adjacent angles ABC, ABD: these
are either two right angles, or are together equal to two right angles. For, if they are equal, then is each of them (def. 10.) a right angle.
But, if not, from the point B draw BE perpendicular to CD (Post. 5.). And because the angle EBD is equal to the two angles, EBA, ABD, to each of these equals add the angle EBC: therefore, (ax. 2.) the two angles EBC, EBD are equal to the three angles EBC, EBĀ, ABD. And in the same manner it may be shewn, that the two angles A B C, ABD, are equal to the same three angles. Therefore, (ax. 1.) the angles A B C, ABD, are together equal to the angles E B C, E B D, that is, to two right angles.
Next, let the straight line A B make with the two straight lines, B C, B D, at the same point B, the adjacent angles ABC, ABD together equal to two right angles: BC, BD shall be in one and the same straight line.
For, let B F be in the same straight line with BC: then, by the first part of the proposition, because AB makes angles with C F upon one side of it, these angles, viz. A BC, A BF, are together equal to two right angles. But ABC, A B D are also equal to two right angles; therefore, (ax. 1.) ABC, A BD together are equal to ABC, ABF together; and, ABC being taken from each of these equals, the angle A B D is equal to ABF (ax. 3.) Therefore B D coincides with BF; that is, it is in the same straight line with B C.
Cor. 2. Any angle of a triangle is. less than two right angles.
PROP. 3. (EUC. i. 15.)
If two straight lines cut one another, the vertical or opposite angles shall be equal.
Let the two straight lines A B, CD, cut one another in the point E: the vertical D angles A E D, BEC, B as also the vertical
angles A EC, BED, shall be equal to one another.
Because the angles AE C, AED are adjacent angles made by the straight line AE with CD, they are (2.) together equal to two right angles; and for the like reason, the angles AEC, CEB, are together equal to two right angles; therefore, (ax. 1.) the angles AEC, AED together are equal to the angles AEC, CEB together. From each of these equals take the angle AE C, and the angle A E D is equal to the angle CE B. (ax. 3.) In the same manner it may be shown that the angles A EC, BE D are equal to one another. Therefore, &c.
Cor. (Euc.i. 15. Cor. 2.) If any number of straight lines pass through the same point, all the angles about that point, (made by each with that next to it,) shall be together equal to four right angles.
PROP. 4. (EUc. i. 4.)
If two triangles have two sides of the one equal to two sides of the other, each to each, and likewise the included angles equal; their other angles shall be equal, each to each, viz. those to which the equal sides are opposite, and the base, or third side, of the one shall be equal to the base, or third side, of the other.
Let ABC, DEF be two triangles, which have the two sides A B, A C, equal to the two sides DE, DF, each to each, viz. A B to D E, and AC to D F, and let them likewise have the angle BAC equal to the angle EDF: their other angles shall be equal, each to each, viz. A B C to DEF, and AC B to DFE, and the base B C shall be equal to the base EF.
For if the triangle B ABC be applied to the triangle DE F, so that the point A may be upon D, and the
straight line A B upon D E, the straight line A C will coincide with D F, because the angle B A C is equal to ED F. Also the point B will coincide with E, because A B is equal to DE, and the point C with F, because A C is equal to DF; and, because the points B, C, coincide with the points E, F, the straight line B C coincides with the straight line E F (ax. 10.), and (ax. 11.) is equal to it; the angle ABC coincides with DEF, and is equal to it; and the angle A C B with the angle D F E, and is equal to it. Therefore, &c.
Cor. The two triangles are equal also as to surface.
It is indifferent which of the two triangles DEF be taken, although in these triangles the side D E lie in opposite directions from DF; viz. to the right of it in the one, and to the left of it in the other. The same may be observed of the next proposition, and of all cases of plane triangles, which are equal in every respect. PROP. 5. (EUC. i. 26, first part of.) If two triangles have two angles of the one equal to two angles of the other, each to each, and likewise the interjacent* sides equal; their other sides shall be equal, each to each, viz. those to which the equal angles are opposite, and the third angle of the one shall be equal to the third angle of the other.
Let ABC, DEF (see the last figure) be two triangles which have the two angles ABC, A C B of the one, equal to the two angles DEF, DFE of the other, each to each, and likewise the side BC equal to the side EF: their other sides shall be equal, each to each, and the third angle BAC shall be equal to the third angle ED F.
For, if the triangle A B C be applied to the triangle D E F, so that the point B may be upon E, and the straight line BC upon EF, the point C will coincide with the point F, because B C is equal to EF. Also the straight line B A will coincide in direction with ED, because the angle CBA is equal to FED, and the straight line CA with FD, because the angle B C A is equal to E FD. But, if two straight lines which cut one another, coincide with other two which cut one another, it is manifest that the points of intersection must likewise coincide. Therefore, the point A coincides with D, and the sides A B, A C, coincide with the sides DE, DF, and are # " Interjacent sides," i. e. sides lying between.
equal to them; and the angle B AC coincides with the angle E D F, and is equal to it (ax. 11.). Therefore, &c.
Cor. The two triangles are equal also as to surface.
PROP. 6. (EUc. i. 5 & 6.)
If two sides of a triangle be equal to one another, the opposite angles shall be likewise equal: and conversely, if two angles of a triangle be equal to one another, the opposite sides shall be likewise equal.
Let A B C be an isosceles triangle, having the side A B equal to the side A C; the angle A C B shall be equal to the angle A B C.
Let the angle B A C be divided into two equal angles by the straight line A D, which meets the base BC in D (Post. 4). Then, because the triangles ADB, ADC have two sides of the one equal to two sides of the other, each to each, and the interjacent angles BAD, CAD equal to one another, their other angles are equal, each to each (4.); therefore the angle A CB is equal to ABC.
Next, let the angle ABC be equal to the angle AC B: the side A C shall be equal to the side A B.
From D, the middle point of BC, and 5.): and, if it do not pass through erect a perpendicular to B C (Post. 3. the vertex A, let this perpendicular, if possible, cut one of the sides as A B in E, and join E C. Then, because the triangles ED B, EDC have two sides of the one equal to two sides of the other, each to each, and the included angles EDB, EDC equal to one another (def. 10.), their other angles are equal, each to each (4.). Therefore the angle ECD is equal to EBD. But EBD or ABD is equal to ACB: therefore the angle ECD is equal to ACB (ax. 1.), the less to the greater, which is impossible. Therefore the perpendicular at D cannot pass otherwise than through the vertex A: and because the triangles ADB, ADC are equal, according to Prop. 4., the side A B is equal to the side A ̊C. Therefore, &c.
also equiangular; and conversely. Cor. 1. Every equilateral triangle is
if the equal sides A B, A C, be proCor. 2. In an isosceles triangle ABC, duced, the angles upon the other side of the base BC will be equal to one another; for, each of them together with