March 1, 1887.] 



KNOW^LEDGE ♦ 



109 



(since a plane through gh, if it pass through kl, must by 

 symmetry pass through sji also). Join kg, lh, ng, and mh. 

 Then the six sides of the figure, klhmng, are all equal, being 

 each equal to half an edge of the octahedron. Moreover, 

 GH, K.Ai, and LN bisect each other in o, by symmetry ; and 

 OG, OK, OL, OH, 051, and ox are all equal, being each equal to 

 half an edge of the octahedron. Hence G, k, l, h, m, n are 

 on the circumfei'ence of a circle about o as centre, and 

 therefore the equal-sided figure gklhiin is a regular 

 hexagon. 



And obviously there are four such hexagonal sections, 

 each section passing through the centre O of the octahedron, 

 and being parallel to two opposite faces. 



Secondly, to fill the hexagonal hole aebcfd with an octa- 

 hedral plug. The figure shows the octahedron of fig. 5, 

 fitted into the hole aebctd ; the letters corresponding in the 

 two figures as in each of the two former cases. 



OUR PUZZLES. 



I HE puzzles which suggest themselves for this 

 month are the following : 



Noting that space can be occupied by a 

 series of equal cubes, which if originally 

 airanged in layers, and each layer in rows, 

 can be shifted in layers and rows : 



Puzzle XXII. Show how to Jill space 

 ■with ffjuitl tttrnh'drons and square-based pyramids, having 

 their triantjniar j'aces equal to those o/ the tetrahedrons ; and 

 determine the movements which can take place among the 

 solids so filling space, without leaving any sjxices vacant. 



Puzzle XXIII. Shoiv how to Jill space with equal tetra- 

 hedrons and with equal octahedrons, having their triangular 

 faces equal to those of the tetrahedrons ; and determine move- 

 ments as in Puzzle XXII. 



Puzzle XXIV. Find a solid, having all its J'aces equal 

 {not, hoivever, one of the regular solids, tvhose J'aces are 

 all regular Jigures), such that space may be filed by equal 

 solids of the kind, no displacements being possible without 

 leaving vacant spaces. 



Those readers who do not care to attack these puzzles in 

 the general form here presented, may take instead the fol- 

 lowing easier problems : 



Puzzle XXII. Arrange four equal regular tetrahedrons, 

 anil six equal square-based pyramids having equilateral 

 triangular faces equal to those of the tetrahedrons, in the 

 form of a square-based pyramid. 



Puzzle XXIII. Arrange eight equal regular tetrahedrons, 

 and six eqital regular octahedrons %vith J'aces equal to those 

 of the tetrahedrons, in thej'orm of a regular octahedron. 



Puzzle XXIV. Draw on a card twelve e<iual and similar 

 diamonds (rhombuses), determining their common shape and 

 their positions on the card so that they may be cut out in 

 one piece, and by suitable bending of diamond towards 

 diamond, each diamond remaining fat, may form the 12 

 faces of a symmetrical solid, having 2-t edges and 1-t angles, 

 6 of the solid angles being each formed by the meeting oj' 

 4 of the smaller plane angles of the diamond faces, ami 8 

 by the meeting of Z of their larger plane angles. 



MINUTE MEASUREMENT. 



[G ontinued from page 80.) 



E have said previously that when extreme 

 delicacy of measurement in an astronomical 

 instrument is essential the micrometer 

 microscope takes the place of the vernier. 

 Now the very essence of the accui-acy of this 

 instrument resides in the absolute truth of 

 the screw which moves its index and cross 

 wires, and, inasmiich as the screw also forms 

 the essential part of eveiy instrument now in use for the 

 measurement of exceedingly minute quantities, it may be as 

 well in beginning this branch of our subject to say a few 

 words as to its principle, and the methods in which a 

 perfectly true screw is produced. If we turn to any popular 

 work in which the so-called "mechanical powers" are 

 described, we shall find the screw defined as a continuous 

 cii'cular wedge or inclined plane, whose outline may 

 be formed by cutting out a wedge-shaped piece of paper 

 and wrapping it round a cylinder. Then will the edge 

 of the paper represent the outline of the screw thread, 

 which ob\-iously preserves a constant angle to the axis of 

 the cylinder ; the angle of the wedge, with a cylinder of 

 unvarying diameter, evidently determining the coarseness 

 or fineness of the screw. We may take it, for our present 

 pui'pose, that a screw must have been originated by very 

 carefully wrapping such a wedge-shaped piece of paper as 

 we have described round a cylinder, and then, with equal 

 care, filing a groove upon it, upon the line traced out by the 

 edge of the paper. The screw being thus made as accurate 

 as possible, would be converted into a " tap " — or cutter — 

 by filing gi'ooves round it parallel to its axis, and with this 

 tap an inside or " female " screw would be cut, which inside 

 screw, or "die" as it is technically called, would in turn 

 become a cutter whence copies of the original screw might 

 be produced in any number. As we are not writing an 

 article on practical mechanics, we need not point out how, 

 by a repetition of these processes, a correction being intro- 

 duced at each stage, the resulting screws come nearer 

 and nearer to perfection, nor describe the elaborate 

 art by which perfectly true screws are now cut with 

 100 threads to the inch. We wUl take their exist- 

 ence for granted, and proceed to describe how they are 

 rendered available for the most minute and delicate 

 measurements. Perhaps as suitable and typiail an applica- 

 tion of the screw as we can select for this purpose may be 

 found in the micrometer microscope, by the aid of which 

 angular deviations are reiid ofli" on the limbs of astronomical 

 circles to the tenth and even hundredth of a second of arc. 

 Let us take one of those used to read the division on the 

 limb of a mural circle, or large Altazimuth instrument, 

 the primary divisions on the circle representing intervals 

 of 5'. Now the microscope itself is a compound one, with 

 an object gla.ss, and that form of eye-piece known as the 

 Eamsden or " positive " one, consisting of two plano-convex 

 lenses with their convexities turned towards each other. 

 Then, as in every other microscope, an image formed by the 

 object glass in its focus will be magnified by the eye-piece, 

 when the focus of the latter is made to coincide with that of 

 the objective by sliding the eye-tube in and out. It will 

 fm-ther be evident that any material object placed accurately 

 at this focal point will be .seen superposed on the optical 

 image formed by the object-glass. Now, in the instru- 

 ment which we are considering, at such focus is placed 

 the spider-line micrometer which is shown in fig .5. 

 Here we see a rectangular frame, moved by the screw sc, 

 whose milled head, m h, is divided into GO equal parts. The 

 circle represents the field of view of the microscope, and the 



