In teaching loci it is helpful to call attention to loci in space (meaning thereby the space of three dimensions), without stopping to prove the proposition involved. Indeed, it is desirable all through plane geometry to refer incidentally to solid geometry. In the mensuration of plane figures, which may be boundaries of solid figures, this is particularly true. It is a great defect in most school courses in geometry that they are entirely confined to two dimensions. Even if solid geometry in the usual sense is not attempted, every occasion should be taken to liberate boys' minds from what becomes the tyranny of paper. Thus the questions: "What is the locus of a point equidistant from two given points; at a constant distance from a given straight line or from a given point?" should be extended to space.[67] The two loci problems usually given at this time, referring to a point equidistant from the extremities of a given line, and to a point equidistant from two intersecting lines, both permit of an interesting extension to three dimensions without any formal proof. It is possible to give other loci at this point, but it is preferable merely to introduce the subject in Book I, reserving the further discussion until after the circle has been studied. It is well, in speaking of loci, to remember that it is entirely proper to speak of the "locus of a point" or the "locus of points." Thus the locus of a _point_ so moving in a plane as constantly to be at a given distance from a fixed point in the plane is a circle. In analytic geometry we usually speak of the locus of a _point_, thinking of the point as being anywhere on the locus. Some teachers of elementary geometry, however, prefer to speak of the locus of _points_, or the locus of _all points_, thus tending to make the language of elementary geometry differ from that of analytic geometry. Since it is a trivial matter of phraseology, it is better to recognize both forms of expression and to let pupils use the two interchangeably. FOOTNOTES: [57] Address at Brussels, August, 1910. [58] For a recent discussion of this general subject, see Professor Hobson on "The Tendencies of Modern Mathematics," in the _Educational Review_, New York, 1910, Vol. XL, p. 524. [59] A more extended list of applications is given later in this work. [60] Ab[=u]'l-'Abb[=a]s al-Fadl ibn H[=a]tim al-Nair[=i]z[=i], so called from his birthplace, Nair[=i]z, was a well-known Arab writer. He died about 922 A.D. He wrote a commentary on Euclid. [61] This illustration, taken from a book in the author's library, appeared in a valuable monograph by W. E. Stark, "Measuring Instruments of Long Ago," published in _School Science and Mathematics_, Vol. X, pp. 48, 126. With others of the same nature it is here reproduced by the courtesy of Principal Stark and of the editors of the journal in which it appeared.