Olaus Magnus Friedrich Henrici

“ We call two planes projective if to every point in one corresponds a point in the other, to every line a line, and to a point in a line a point in the corresponding line, in such a manner that the cross-ratio of four points in a line, or of four rays in a pencil, is equal to the cross-ratio of the corresponding points or rays. ”

“ The moment of inertia of a plane figure does not change if the figure undergoes a shear in the direction of the axis with regard to which the moment has been taken.
If a figure has an axis of skew-symmetry, then this axis and' the conjugate direction are conjugate diameters of the momental ellipse for every point in the axis.
If a jigure has an axis of symmetry, then this is an axis of the momental ellipse for every point in it. The truth of the last propositions follows at once from the fact that the product of inertia for the lines in question vanishes. ”

“ The accuracy and durability of any machine depend to a great extent on the manner in which the carrying mechanism is constructed. Besides, no wheel must be capable of moving in any other way than that required; hence every part must be locked and be released only when required to move. Further, any disk must carry to the next only after the carrying to itself has been completed. If all were to carry at the same time a considerable force would be required to turn the handle, and serious strains would be introduced. ”

“ The point of contact of the wheel r or any point of the carriage C will therefore always move in a direction making an angle φ with the axis of x, whilst it moves in the x-direction through the same distance as the point B on the y-curve—that is to say, it will trace out the integral curve required, and so will any point rigidly connected with the carriage C. A pen P attached to this carriage will therefore draw the integral curve. Instead of moving B along the y-curve, a tracer T fixed to the carriage A is guided along it. ”

“ If in two perspective planes one pair of corresponding points be harmonic conjugates with regard to the centre and the point where the line joining them cuts the axis, then every pair of corresponding points has this property and the planes are in involution. ”

“ At the same time five points A, B, C, A', B on the conic have been found, so that the conic is completely known which determines on the line s the given involution. Hence-Through three points wecan always draw one conic, and only one, which determines on a given line a given involution, all the same whether the involution has real, coincident or invisible foci. In the last case the theorem may now also be stated thus:- It is always possible to draw a conic which passes through three given real points and through two invisible points which any other conic has in common with a line. ”

“ We have seen (§ 21) that a conic may always be projected into itself by taking any point S as centre and its polar s as axis of projection, corresponding points being those in which a line through S cuts the conic. If then (fig. 9) A, A' and B, B' are pairs of corresponding points so that the lines AA' and BB' pass through S, then the lines AB and A'B', as corresponding lines, will meet at a point R on the axis, and the lines AB' and A'B will meet at another point R' on the axis. ”

“ Practically all calculating machines contain a “counting work,” a series of “figure disks” consisting in the original form of horizontal circular disks (fig. 1) , on which the figures 0, 1, 2, to 9 are marked. Each disk can turn about its vertical axis, and is covered by a fixed plate with a hole or “window” in it through which one figure can be seen. On turning the disk through one-tenth of a revolution this figure will be changed into the next higher or lower. Such turning may be called a “step,” positive Addition machines. if the next higher and negative if the next lower figure appears. ”

“ The wheel W rolls on the cone to an amount again proportional to dx, and also proportional to y, its distance from V. Hence the roll of the wheel is proportional to the area ydx described by the rod QT. As T is moved from A to B along the curve the roll of the wheel will therefore be proportional to the area AA′B′B. If the curve is closed, and the tracer moved round it, the roll will measure the area independent of the position of the axis OX, as will be seen by drawing a figure. The cone may with advantage be replaced by a horizontal disk, with its centre at V ”

“ FF is a figure disk with a conical wheel A on its axis. In the covering plate HK is the window W. A stepped cylinder is shown at B. The axis Z, which runs along the whole machine, is turned by a handle, and itself turns the cylinder B by aid of conical wheels. Above this cylinder lies an axis EE with square section along which a wheel D can be moved. The same axis carries at E′ a pair of conical wheels C and C′, which can also slide on the axis so that either can be made to drive the A-wheel. ”

“ In the Brunsviga, the figure disks are all mounted on a common horizontal axis, the figures being placed on the rim. On the side of each disk and rigidly connected with it lies its A-wheel with which it can turn independent of the others. The B-wheels, all fixed on another horizontal axis, gear directly on the A-wheels. By an ingenious contrivance the teeth are made to appear from out of the rim to any desired number. The carrying mechanism, too, is different, and so arranged that the handle can be turned either way, no special setting being required for subtraction or division. ”

“ If two similar figures are perspective without being in the same plane, their planes must be parallel as the axis is at infinity. Hence-Any plane figure is projected from any centre to a parallel plane into a similar figure. ”

“ If we turn round and go in the opposite sense, then the area will be to the left if it was first to the right, and vice versa. If we give the boundary a definite sense, and go round in this sense, then the area is said to be either of the one or of the other sense according as the area is to the right or to the left. ”

“ The displacement z, which necessarily must be small, can be enlarged by aid of a lever OT′. To regulate the displacements y of the points C (fig. 24) each spring is attached to a lever EC, fulcrum E. To this again a long rod FG is fixed by aid of a joint at F. The lower end of this rod rests on another lever GP, fulcrum N, at a changeable distance y ′′ = NG from N. The elongation y of any spring s can thus be produced by a motion of P. If P be raised through a distance y ′, then the displacement y of C will be proportional to y ′y ′′ ”

“ Projective Properties.-The properties of the projection of a figure depend partly on the relative position of the planes-of the figures and the centre of projection, but principally on the properties of the given figure. Points in a line are projected into points in a line, harmonic points into harmonic points, a conic into a conic ”

“ CALCULATING MACHINES. Instruments for the mechanical performance of numerical calculations, have in modern times come into ever-increasing use, not merely for dealing with large masses of figures in banks, insurance offices, &c., but also, as cash registers, for use on the counters of retail shops. ”

“ One guiding of the tracer over the curve gives then at once the ten coefficients An and Bn for n = 1 to 5.
All the calculating machines and integrators considered so far have been kinematic. We have now to describe a most remarkable instrument based on the equilibrium of a rigid body under the action of springs. The body itself for rigidity’s sake is made a hollow Michelson and Stratton analyzer. cylinder H, shown in fig. 24 in end view. It can turn about its axis, being supported on knife-edges O. To it springs are attached at the prolongation of a horizontal diameter ”

“ This is the principle of Babbage’s difference machine. The clock dials have to be replaced by a series of dials as in the arithmometers described, and an arrangement has to be made to drive the whole by turning one handle by hand or some other power. ”

“ Multiplying by 817 requires in the Thomas 16 turns of the handle, but in the Steiger-Egli only 3 turns, with 3 settings of the lever H. If the lever H is set to 1 we have a simple addition machine like the Thomas or the Brunsviga. The inventors state that the product of two 8-figure numbers can be got in 6–7 seconds, the quotient of a 6-figure number by one of 3 figures in the same time, while the square root to 5 places of a 9-figure number requires 18 seconds.
Machines of far greater powers than the arithmometers mentioned have been invented by Babbage and by Scheutz. ”

“ Against these gear two disks, B and C, with radii a; their axes are fixed in the carriage. From the disk A extends to the left a rod OT of length l, on which a recording wheel W is mounted. The disks B and C have also recording wheels, W1 and W2, the axis of W1 being perpendicular, that of W2 parallel to OT. If now T is guided round a figure F, O will move to and fro in a straight line. This part is therefore a simple planimeter, in which the one end of the arm moves in a straight line instead of in a circular arc. Consequently, the “roll” of W will record the area of the figure. ”

“ To adapt an addition machine, as described, to rapid multiplication the turnings of the separate figure disks are Modified addition machines.replaced by one motion, commonly the turning of a handle. As, however, the different disks have to be turned through different steps, a contrivance has to be inserted which can be “set” in such a way that by one turn of the handle each disk is moved through a number of steps equal to the number of units which is to be added on that disk. ”

“ If the whole block of templets be now pushed under the points P and if the drawing-board be moved at the same rate, then the pen T will draw the curve Y = F (ξ) . The instrument now is an integraph giving the value of a definite integral as function of a variable parameter.
Having thus shown how the lever with its springs can be made to serve a variety of purposes, we return to the description of the actual instrument constructed. The machine serves first of all to sum up a series of harmonic motions or to draw the curve
The motion of the points P1P2 ... ”

“ It may be noticed at once that the roll of the wheel gives at every moment the area A′ATQ. It will therefore allow of registering a set of values of ydx for any values of x, and thus of tabulating the values of any indefinite integral. In this it differs from Amsler’s planimeter. ”

“ In this case, if we take the triangle together with the line at infinity, we get as the projection a triangle ABC, and some other line j which cuts the sides a, b, c of the triangle in the points Al, Bl, Cl. ”

“ By aid of a lever H the block can be raised or lowered so that any row of the block comes to the level of the racks, the units-rods being opposite the ends of the racks.
The action of the machine will be understood by considering an example. ”

“ They are all driven by one handle. These disks take the place of the templets described before. The distances NG are made equal to the amplitudes a1, a2, a3, ... The drawing-board, moved forward by the turning of the handle, now receives a curve of which (2) is the equation. If all excentrics are turned through a right angle a sine-series can be added up.
It is a remarkable fact that the same machine can be used as a harmonic analyser of a given curve. ”

“ These springs are supposed to follow Hooke’s law. If the elongation beyond the natural length of a spring is λ, the force asserted by it is p = kλ. Let for the position of equilibrium l, L be respectively the elongation of a small and the large spring, k, K their constants, then
Fig. ”

“ This proves the last part of the theorem in 5.
§ 8. The plane -/r' may be turned either in the sense indicated by the arrow at Z or in the opposite sense till -rr' falls into -rr. In the first case we get a figure like fig. 3; i' andj will be on the same side of the axis, and on this side will also lie the centreS ”