The instruments to which references are made in Chapter IV. as having come into use in some of our leading mercantile shipyards by which the calculations undertaken there are rendered greatly more simple, and are more expeditiously made, seem not to be generally known amongst shipbuilders, and as they undoubtedly save much of the labour and time of calculation, without any sacrifice of accuracy, illustrations of them are here given, together with brief notes of their construction and use. For anything, however, like a satisfactory account of the mathematical principles on which these several instruments are based, readers must consult the authoritative sources to which references will be made.
Assuming that the reader appreciates the advantages of shortened calculation, due to the slide rule, or the use of logarithms, the first instrument that may be noticed is one embodying an application of the principle of the slide rule in a remarkably handy and compact form. This is the calculating slide rule invented by Professor Fuller, of Queen’s College, Belfast, equivalent to a straight slide rule 83 feet 4 inches long, or a circular rule 13 feet 3 inches in diameter. From the illustration given it may be seen that the rule consists of a cylinder which can be moved up and down upon, and turned round, an axis, which is held by a handle. Upon this cylinder is wound spirally a single logarithmic scale. Fixed to the handle of the instrument is an index. Two other indices, whose distance apart is the axial length of the complete spiral, are fixed to an inner cylinder, which slides in like a telescope tube, and thus enables the operator to place these indices inany required position relative to the outer cylinder containing the logarithmic scale. Two stops—one on the fixed and the other on the outer or movable cylinder—are so placed that when they are brought in contact the index points to the commencement of the scale.
FIG. 24.Logarithmic calculatorFULLER’S RULE.
FULLER’S RULE.
Regarding the manner of using the instrument a few general notes may be given. As in the ordinary slide rule the operations of multiplication and division are performed by the addition or subtraction of the parts of the scale that represent in length the logarithm of the numbers involved in the operations.
For example, suppose the following calculation is to be worked out
(6248 × 5936 × 4217)(7963 × 4851)= 4049
To do this in the ordinary way would keep the smartest arithmetician busy for a considerable time, whereas by means of the instrument under notice the result is attained in little over one minute’s time. The motions in the operation are as follows:—Hold the rule by the handle in one hand and move the scale cylinder by the other until the number 6248 is opposite the index attached to the handle portion. Now, move the inner cylinder (by the top) until one or other of the indices (according to the distance of the number from the bottom of the instrument) on the index arm is opposite the number 7963. The scale cylinder is again moved till the number 5936 is opposite one of the indices just referred to, and the inner cylinder carrying the index arm is then moved till one or other of the indices is opposite 4851. Finally, the scale cylinder is moved till the number 4217 is opposite one of the indices on the arm; and the result of the whole operation—4049—is found opposite the index first-mentioned,i.e., that attached to the handle portion of the instrument.
It may be further explained that the sliding of the scale cylinder until the new number is opposite the index point really involves two operations: one sliding it till the end of the scale is opposite the index point—which subtracts the logarithm of the divisor; and the other sliding it till the next multiplier is opposite the index point—which adds its logarithm to the previous result. Hence, when the operations end with division the scale cylinder must be moved till the end of the scale is opposite the index point.
The second scientific instrument to be noticed is the Polar Planimeter, invented by M. J. Amsler-Laffon, Schaffhausen, Switzerland, the object of which is to find the area of any figure by simply tracing the outline with a pointer, the instrument—of which the pointer is a part—doing all the rest; the results read off from it having to undergo only a very simple and elementary calculation to attain the desired result.
FIG. 25.Machine to calculate areaAMSLER’S POLAR PLANIMETER—(FIXED SCALE).
AMSLER’S POLAR PLANIMETER—(FIXED SCALE).
Planimeters are made of several forms, the two kinds illustrated by Figs. 25 and 26 being the most usual.[34]The planimeter shown by Fig. 25 represents the instrument as made to one scale only, for square inches of actual measurement. Byits means the areas of, say, cross sections of ship’s hull can be ascertained in an extremely short time and with almost perfect accuracy, the readings taken from the instrument having simply to be multiplied by a multiplier consisting of the square of the number of units to the inch, corresponding to the scale on which the sections are drawn, as 4 for ½-inch scale, 16 for ¼-inch, 64 for ⅛-inch, etc.
FIG. 26.Machine to calculate areaAMSLER’S PLANIMETER—(VARIOUS SCALES).
AMSLER’S PLANIMETER—(VARIOUS SCALES).
The Planimeter shown by Fig. 26 is the instrument in a form adaptable to various scales, but does not possess any very marked advantages over the simpler form for the purposes of the naval architect or marine engineer, so that notice of it must be brief. In this form of the instrument the unit can be changed by altering the length of the arm which carries the tracer to any of the scales for which the instrument may be made available, and which are found divided upon the variable arm. The scales which are usually provided for are as follows:—
Describing the simple planimeter more in detail, and referring to Fig. 25, it may be said the outline of the figure to be dealt with is travelled round by a pointer attached to a barmoving on a vertical axis carried by another bar, which latter turns on a needle point slightly pressed into the drawing surface. The bar with the pointer is provided with a revolving drum having a graduated circumference and a disc counting its revolutions. The drum is divided into 100 parts, reading into a vernier, which gives the reading of the drum’s revolution to the1/1000part of its circumference. Upon the same axis as the drum an endless screw is cut, working into a worm wheel of ten teeth connected with the counting disc, which records the revolutions of the drum.
To use the planimeter, place the instrument upon the paper so that the tracing point, roller, and needle point, all touch the surface at any convenient position. Press the needle point down gently, so that it just enters the paper, and place the small weight supplied with the instrument over it. Make a mark at any part of the outline of the figure to be computed, and set the tracing point to it. Before commencing read off the counting wheel and the index roller. Suppose the counting wheel marks 2, the roller index 91, and the vernier 5, then, the unit in this case being 10 sq. ins., write this down 29·15 (for the proportional or variable-scale planimeter this reading would be 2·915.) Follow with the tracing point exactly the outline of the figure to be measured in the direction of the movement of the hands of a watch, until you arrive at the starting point; now read the instrument. Suppose this reading to be 47·67, then by deducting the first reading (29·15) the remainder (18·52) indicates that the measured area contains 18·52 units—i.e., square inches—which is the final result, so far as the instrument is concerned. To obtain the actual area in feet, however, this result must be multiplied by the number before explained corresponding to the scale on which the figure that has been measured is drawn.[35]Assuming the scale to have been ¼-inch per foot, then 18·52 inches multiplied by 16—the appropriate multiplier for that scale—gives 296·32 square feet, the exact area.
Several important points remain to be noticed in connection with the use of the instrument. As a rule, the areas to be measured in connection with ship designing are on a small scale, and the fixed or needle point about which the instrument moves can always be placedoutsidethe figure measured, in which case the process remains as above stated. It should be mentioned, however, that by placing the needle pointinsidethe figure, in such a position as to enable the operator to follow its contour a larger figure can be measured at one operation—the reading, however, being less than the true area by a constant number which varies slightly with the construction of each instrument, and which is found engraved on the small weight already referred to (on the top of the bar in the proportional planimeter). Adding this constant number to any reading taken by the instrument placed as described, gives the true area.
The counting disc may go through more than one revolution forwards or backwards. If the needle point beoutsidethe figure traversed the counting disc can only moveforwards(as 9, 0, 1, 2, &c.): that is, provided the figure has been traced in the manner directed—in the direction of the hands of a watch. Then as many times as the zero mark passes the index line add 10·000 to thesecondreading. If the needle point beinsidethe figure, the disc can move either forwards or backwards. If moving backwards, as 2, 1, 0, 9, &c., then add 10·000 to thefirstreading.
Before passing from the subject of the planimeter it may be both interesting and useful to give an example of a calculation involving its use. Subjoined is a specimen displacement and longitudinal centre of buoyancy calculation, and any one familiar with the prodigious array of columns and figures pertaining to a “displacement sheet” of the ordinary kind cannot fail to appreciate the advantages of the specimen, both with respect to simplicity of arrangement and curtailment of theamount of calculation ordinarily involved:—
EXAMPLE OF SHIP DISPLACEMENT, WORKED OUT BY PLANIMETER.
The integrator, another and still more ingenious instrument, by M. J. Amsler-Laffon, was invented theoretically shortly afterthe planimeter justdescribed (in the year 1855), but was first constructed for practical use in the year 1867, the first instrument made being exhibited in the Paris International Exhibition in the year named. It was not introduced into England till the year 1878, and although adapted for other uses than thoseinvolved in scientific calculations connected with shipbuilding it was in this connection that attention was first seriously directed towards it. In 1880 the late Mr C. W. Merrifield described the instrument, and traced the mathematical principles upon which it is based, before the Institution of Naval Architects, and in 1882, before the same body, Mr J. H. Biles, naval architect for the firm of Messrs J. & G. Thomson, called attention to the usefulness of the instrument in stability investigations, showing by specimen calculations and other particulars its great adaptability to this class of work, even in the hands of youthful and untrained operators. A still more recent and exhaustive paper devoted to the claims of the integrator upon naval architects was read before the same Institution by Dr A. Amsler, the son of the inventor, at its last meeting. This paper was chiefly concerned with demonstrating the advantages of the integrator in respect of time saved, as well as in respect of its great accuracy.
FIG. 27.Machine to calculate area and momentsAMSLER’S MECHANICAL INTEGRATOR.
AMSLER’S MECHANICAL INTEGRATOR.
The object of the integrator is to find at one operation the area, the statical moment, and the moment of inertia of any closed curve or figure by simply tracing out the curve with apointer, the results being read off directly from the instrument, as in the case of the planimeter, and with a correspondingly small amount of after calculation. As shown by Fig. 25, the essential parts of the integrator are a railL, having groove with which to guide the wheelspandqof a carriage provided with rollersD1D2D3moving on the surface of the drawing. The contour of the figure to be dealt with is traced—in the direction of the movement of the hands of a watch—by the pointerF, this pointer being attached to an arm moving on the vertical centre of the instrument while the whole mechanism runs to and fro on the railL. Under these conditions the rollersD1D2D3execute movements partly rolling, partly sliding, and by readings taken from the divisions engraved upon their circumferences at the beginning and the end of the whole movement, together with simple arithmetical processes, the nature of which may be inferred from the explanations given of the planimeter readings, the three quantities sought are arrived at.
In a valuable appendix to the paper read by Dr Amsler, before the Institution of Naval Architects, specimen sheets are given of several calculations, of a vessel of about 4000 tons, the forms in which the figures are entered being so arranged as to avoid all unnecessary trouble in measuring and calculating, and to contain at the same time a check on the results. The accuracy and the speed of working depend, of course, to a considerable extent on the person using the integrator, but as showing what can be obtained with the instrument after some practice, the specimens given in the paper referred to are certainly remarkable. For the calculations of the data necessary for the construction of the curves of displacement and vertical position of centre of buoyancy, the complete integrator and arithmetical work took only two hours; for the data requisite for the curve of displacement per inch immersion, and transverse metacentre one hour was taken; and for the complete calculation, affording data to construct a stability curve, the time taken was only eight hours. A similar calculation done in the ordinary arithmetical method, and giving results far less reliable, would have taken as many days. All the work, it should be added, wasdone without the aid of an assistant. Amongst other calculations besides displacement and stability in connection with which the integrator is greatly advantageous, are those concerned with the strength of vessels and with the longitudinal strains to which they are subject at sea through unequal distributions of weight and buoyancy, already fully referred to in the chapter on scientific progress.
BENNETT & THOMSON, PRINTERS.