4. If P'Q'R'S' be another parallelogram described as in Ex. 1, the intersections of PQ, P'Q', &c. shall be in the angular points of a parallelogram EFGH constructed from PQRS as P'Q'R'S' is constructed from ABCD.

5. The quadrilateral formed by bisecting the sides of a quadrilateral and joining the successive points of bisection is a parallelogram, with the same mean point.

6. If the same be true of any other equable division such as trisection, the original quadrilateral is a parallelogram.

7. If any line pass through the mean point of a number of points, the sum of the perpendiculars on this line from the different points, measured in the same direction, is zero.

8. From a point E in the common base AB of the two triangles ABC, ABD, straight lines are drawn parallel to AC, AD, meeting BC, BD at F, G; shew that FG is parallel to CD.

9. From any point in the base of a triangle, straight lines are drawn parallel to the sides: shew that the intersections of the diagonals of every parallelogram so formed lie in a straight line.

10. If the sides of a triangle be produced, the bisectors of the external angles meet the opposite sides in three points which lie in a straight line.

11. If straight lines bisect the interior and exterior angles at A of the triangle ABC in D and E respectively; prove that BD, BC, BE form an harmonical progression.

12.

The diagonals of a parallelepiped bisect one another.

13. The mean point of a tetrahedron is the mean point of the tetrahedron formed by joining the mean points of the triangular faces; and also those of the edges.

14. If the figure of Ex. 11, Art. 7, be that of a gauche quadrilateral (a term employed by Chasles to signify that the triangles

AOD, BOD are not in the same plane), the lines QP, DO, RS will meet in a point, provided

15. If through any point within the triangle ABC, three straight lines MN, PQ, RS be drawn respectively parallel to the sides AB, AC, BC; then will

16. ABCD is a parallelogram; E, the point of bisection of AB; prove that AC, DE being joined will trisect each other.

17. ABCD is a parallelogram; PQ any line parallel to CD ; PD, QC meet in S, PA, QB in R; prove that AD is parallel to RS.

CHAPTER III.

VECTOR MULTIPLICATION AND DIVISION.

15. We trust we have made the reader understand by what we stated in our Introductory Chapter, that, whilst we retain for 'multiplication' all its old properties, so far as it relates to ordinary algebraical quantities, we are at liberty to attach to it any signification we please when we speak of the multiplication of a vector by or into another vector. Of course the interpretation of our results will depend on the definition, and may in some points differ from the interpretation of the results of multiplication of numerical quantities.

It is necessary to start with one limitation. Whereas in Algebra we are accustomed to use at random the phrases 'multiply by' and 'multiply into' as tantamount to the same thing, it is now impossible to do so. We must select one to the exclusion of the other. The phrase selected is 'multiply into'; thus we shall understand that the first written symbol in a sequence is the operator on that which follows: in other words that aß shall read 'a into ẞ', and denote a operating on B.

.

16. As in the Cartesian Geometry, so here we indicate the position of a point in space by its relation to three axes, mutually at right angles, which we designate the axes of x, y, and z respectively. For graphic representation the axes of x and y are drawn in the plane of the paper whilst that of z being perpendicular to that plane is drawn in perspective only. As in ordinary

X

geometry we assume that when vectors measured forwards are represented by positive symbols, vectors measured backwards will be represented by the corresponding negative symbols. In the figure before us, the positive directions are forwards, upwards and outwards; the corresponding negative directions, backwards, downwards and inwards.

With respect to vector rotation we assume that, looked at in perspective in the figure before us, it is negative when in the direction of the motion of the hands of a watch, positive when in the contrary direction. In other words, we assume, as is done in modern works on Dynamics, that rotation is positive when it takes place from y to z, z to x, x to y: negative when it takes place in the contrary directions (see Tait, Art. 65).

Unit vectors at right angles to each other.

17. DEFINITION. If i, j, k be unit vectors along Ox, Oy, Oz respectively, the result of the multiplication of i into j or ij is defined to be the turning of j through a right angle in the plane perpendicular to i and in the positive direction; in other words, the operation of i on j turns it round so as to make it coincide with k; and therefore briefly ij=k.

To be consistent it is requisite to admit that if i instead of operating on j had operated on any other unit vector perpendicular to i in the plane of yz, it would have turned it through a right angle in the same direction, so that ik can be nothing else than −j. Extending to other unit vectors the definition which we have illustrated by referring to i, it is evident that j operating on k must bring it round to i, or jk = i.

Again, always remembering that the positive directions of rotation are y to z, z to x, x to y, we must have ki=j.

18. As we have stated, we retain in connection with this definition the old laws of numerical multiplication, whenever numerical quantities are mixed up with vector operations; thus 2i. 3j6ij. Further, there can be no reason whatever, but the contrary, why the laws of addition and subtraction should undergo

T. Q.

3

any modification when the operations are subject to this new definition; we must clearly have

i (j + k) = ij + ik.

Finally, as we are to regard the operations of this new definition as operations of multiplication-magnitude and motion of rotation being united in one vector symbol as multiplier, just as magnitude and motion of translation were united in one vector symbol in the last chapter-we are bound to retain all the laws of algebraic multiplication so far as they do not give results inconsistent with each other. In no other way can the conclusions be made to compare with those deduced from the corresponding operations in the previous science. Thus we retain what Sir William Hamilton terms the associative law of multiplication: the law which assumes that it is indifferent in what way operations are grouped, provided the order be not changed; the law which makes it indifferent whether we consider abc to be a × bc or ab × c. This law is assumed to be applicable to multiplication in its new aspect (for example that ÿjk=ÿj. k), and being assumed it limits the science to certain boundaries, and, along with other assumed laws, furnishes the key to the interpretation of results.

The law is by no means a necessary law. Some new forms of the science may possibly modify it hereafter. In the meantime the assumption of the law fixes the limits of the science.

The commutative law of multiplication under which order may be deranged, which is assumed as the groundwork of common algebra (we say assumed advisedly) is now no longer tenable. And this being the case it is found that the science of Quaternions breaks down one of the barriers imposed by this law and expands itself into a new field.

ij is not equal to ji, it is clearly impossible it should be.

A simple inspection of the figure, and a moment's consideration of the definition, will make this plain. The definition imposes on i as an operator on j the duty of turning j through a right angle as if by a left-handed turn with a cork-screw handle, thus throwing jup from the plane xy; when, on the other hand, j is the operator