US157239A - Improvement in philosophical instruments or estimators - Google Patents

Description

ATTUBNEYS.

M. STAPFF.

Patented Nov.24,1874.

Philosophical Instruments or Estimators."

UNITED STATES PATENT OEEICE.

FREDRIG MAURICE STAPFF, OF STOCKHOLM, SWEDEN.

IMPROVEMENT IN PHILOSGPHICAL INSTRUMENTS OR ESTIMATORS.

Specilication forming p art of Letters Patent N0. 157,239, dated November 24, 1874; application tiled October 4, 1873.

To all whom it may concern:

' Be it known that I, FEEDnrc MAURICE STAPFF, of the city of Stockholm, in the Kingdoln of Sweden, have invented a new and Improved Estimator, of which the following is a specification:

The estimator is a sliding rule, by which the volume oi prismatoidal bodies is calculated mechanically. As mo'st of the einbankments, ditches, cuts, 85o., occurring in the construe tion of railroads, canals, fortiiications, Ste., possess prismatoidal shape, the estimator has .the power to abridge and facilitate the important but tedious task of computing` the quantities in earthwork. This task is facilitated by tables, which (in America) are based upon the prismatoidal formula, viz:

rf e E HIL: a {TID-at Making use of those two fundamental i'ormulas in the construction of my instrument, I have found it proper to transform the first one of them as follows:

If the volume sought has to be expressed by cubic yards, While the sizes of prisinatoid are measured by foot, this formula reads:

rI he following description, with drawing, refers to an estimator constructed in accordance with this last formula. But it should. be remembered that the arrangement of my instrument by no means needs to be changed, and only three ot' its eight scales have to be modi- :Iied if the proportion between unity of length and unity of volume be another one than that of foot to cubic yard.

Vihen compared with the tables now in use, the estimator offers the following advantages: It may be applied with equal ease, whatever (within the extent of the estimator-scales) the values of B and a entering the calculation may be, while the said tables are made up but for certain oi'teuoccurring values oi' B and a it allows to nish the calculation with more exactness and speed, and with less mental ei'- i'ort, than the tables do; it may be taken out to the field in a great-coat pocket, while an equivalent complete collection of tables would represent a little library; iinally, one estimator worth about $20 will, to each railroadoffice, save the expense for one or more assist anis.

On the drawing adjoined to this description, and representing an estimator in reduced scale, the diiierent scales forming essential parts of the instrument are only generally indicated.

rlihe estimator is composed of they rule b, in which two slides, a and c, move, one above the other, parallel to the axis of the rule. The lower slide a is provided with a diagram of curves, by means of which the average heights are found. The upper slide c carries iive parallel scales, (numbered 3 to 7,) and the rule b carries two scales (numbered l and 2) at the one side, and one scale (numbered S) at the other side of slidec. Of those scales, No. 7 and No. 8 are used for multiplications and arithmctical operations of similar nature. rllhe distances from the starting-points of those scales to their did'erent graduated lines are proportional to the Briggs logarithms of the numerical values expressed by the respective graduated lines. I prefer ina-king the graduation of one of those scales to progress from the left to the right, that of the other one in opposite direction.

For the special purpose of the estimator it is suilicient if the multiplication-scales represent the logarithms of numerical values between i).05 and 1000; but if the same scales should be used for multiplications, Cto., of

greater or smaller values, it is but necessary to suppose the graduated lines to mean 1m or l, or ten times or one hundred times, the values printed on the scales, and then to operate in the ordiiiaiy way. Besides, the extent of the graduation on those scales may be varied at pleasure.

For pcrforniin g iiiiiltiplicatioiis by means of scales 7th and Sth, the slide c should be drawn out until the graduated line indicating one of the factors on one of those scales is opposite that graduated line on the other scale by which the other factor is indicated. rllie product then is read on either of the two scales opposite the I line of the other one.

For working divisions the slide c should be drawn until the I line on one of the scales is in opposition with that graduated liiie on the other scale by which the dividend is indicated. Bead then the quotientfrom any of the scales opposite that graduated line on the other scale by which the division is indicated.

Square roots are extracted by moving the slide c until the I line on one of the scales is opposite to the graduated line on the other scale indicating the power. The root is then indicated by those two graduated lilies on both scales which coincide and express identical numerical values. A L-sliaped groove, 7.', ruiming along scale Sth in rule c, contains two buttons, B and n, which, in this groove, are movable to and fro. rlliese bnttoiis are made use of for indicating factors to be multiplied by B or n.

The graduation of scale 2d (on rule D) and of scale 3d (on slide c) is identical and uniform on both scales, the intervals between equivalent graduated liiies being equal, but one of the scales (2d) is ciphered from the left to the right, the other one (3d) in opposite direction.

rlhe extent of the cipher series expressed on the addition-scales depends upon the extent of the work to be performed by the estimator. If the sum ot' the heights of both terminal cross-sections (H+7z) does not exceed 100', it is sutlicient to express by each of the additioii-scales the cipher series 0 to l0() with subdivisions. c

For summing up two numerical values,f, i', H, and 7i, the slide c should be dra-wn until the graduated line on scale 2d, expressing' one of the sums coincides with the graduated line by which on scale 3d the other sinn is expressed. rlhe sum then niay be read from either scale opposite the Zero-line of the other one. The zero-line of scale 3d being indicated by the edge of iiidex-plate d, (on slide 0,) is commonly used to read the sums by.

If two values have to be summed the sinn of which is greater than 100 and less than 200, the operation is quite the saine as here de. scribed; but to the value read from scale 3d, opposite the 100 line of scale 2d, 100 must be added. For suniniin any two values with the assistance of the addition-scales, it is but necessary to understand the graduated lines to mean l0 or 100 times tlie values printed on the scales, and then to operate as described hereabove.

For effectingsubtractioiis by means of the estiniators addition-scales, the slide c should be drawn until the index-edge d meets with that graduated line on scale 2d by which the subtrahend is expressed. Bead the rest from scale 3d or 2d, opposite that graduated line on 2d or 8d by which the subtractor is expressed.

By the scale 6th on slide c the function is expressed. The graduation of this scale is unii'orin, the intervals between equivalent degrees being 5L times wider than those on scale 2d, where the sums H-l-i are produced as described above. The scale begins at the edge of iiidex-plate d, exactly below the Zero-line of scale 3d, proceeds from the right to the lelt, is ciphered in the same direction, and read by the edge of index-plate g, which is fixed on rule Z), exactly the zero-line of scale 2d. Whatever the position of slide c (inside the rule Z1) may be, the distance from the Zero-line of scale 2d to the index-plate d always must be equal to the distance between the index-plates d and g. Of course, if any snm (H4-h) produced by lielp of the additionseales 2d and 3d is read from scale 2d by the edge of index d, synchronously the function H-l-L 5i of index-plate g.

The subdivisions of scale 6th may be more or less extended, but, on estimators for coinn'ion use, I prefer to express on this scale directly by graduated lines ,-166 c. yard.

By the scale 5th on slide c the function may be read from scale 6th by the edge l is expressed. This scale proceeds from the edge of index-plate g; the distance from d to g, by which is indicated,

always being equal to the distance from the zero-point of scale 2d to d, by which (H-l-h) is indicated. You will read from scale 5th, by

T-S if you synchronously may iid IfI-i-L on scale 2d by index d.

The scale is constructed by the formula H-l-h: v/lllr, H-i-L meaning the distance from d to the graduated line to be setout; a7, the value to be expressed by that graduated line. rIlie subdivision ot' this scale may be extended more or less; but, on estimators ot' common size, I prefer to express directly, by graduated lines, l e. yards between 0 and 25; c. yards between 25 and the end of scale.

By the scale 4th the function is expressed. This scale begins at the edge of plate d, progresses from the right to thc left,

index g, the function and is read at the. edge of index-plate i, which may be drawn to and fro in the groove 7c, between the scales 1st and 2d. By means of the scale 1st (on rule b) this index can be placed so that the distance from it to edge of indexplate Zexpresses the difference H-K-t, while the sum H+ h is synchronously expressed by the distance from g to d. Scale lst starts exactly above the zero-point of scale 2d, thence pro gressing to the right. Its graduation is a uniform one7 the intervals between equivalent graduated lilies being twice as wide as on scale 2d. Of course, if by index z' any value of 7a is indicated on scale lst, the value 2h will be synchronously indicated by the same index on scale 2d. If slide c is in such a position that the edge of plate d coincides with any graduated line on scale 2d expressing the value H-l-k, and at the same time the index t' is in such a position that by it may be read h from scale lst, or 2h from scale 2d, then the value H+h-2r=Hh will be indicated by H-h 2 M bythe saine index on scale 4th.

Scale 4th is constructed by the formula II-L: VMzIS 1,/ meaning the numerindex on scale 3d, and the function ical value to be expressed by any graduated H I The value is read from scale 6th by index g, and marked on scale 8th by the button B.

H 2 The value is read from scale 5th as read from scale 4th by index z'. The snm of both is marked on scale 8th by the button u.

The multiplication B (H 7L) is done by drawing slide c until the graduated line on scale 7th which expresses the value B coincides with the indicating-edge of button B. The product read from either of the scales 7th or 3th, opposite the I line of the other scale, is marked on scale 2d byindex t'. The multipli# by index g, and added to the value VII-l-h 2 Eli-.7L 2 catlonux T8 w: :I is done by drawing slide c until the graduated line (marked by index z' on scale 2d) by last-found product, a

The sum read from scale 2d by edge of index d v expresses, by cubic yards, the volume of one running foot of the prismatoid. Let the length L be 100, as commonly in America, where the distance between two station-points is 100', the volume of the whole prismatoid is found simply by placing the decimal-mark of the sum next after the second fractional cipher to the right hand; but, if the prismatoid extending between intermediate or plus points be shorter than 100', the multiplication L x sum is easiest done with the assistance of the multiplication-scales 7th and Sth.

Lethbe 25'; H :.55; n 1%;13 22.5; L 100. After having marked 25 by index t on scale 1st, draw slide c until the graduated line 25 on scale 2d coincides with the graduated line 55 on scale 3d. Bead from scale 6th, by the edge of index g, 1.48, and mark this value by index B on scale 8th. Read from scale 4th, by the index z', 2.78, and from scale 5th, by the edge of index g, 59.26. Add both values together and mark the sum, 62.04, by index n on scale 8th. Draw slide c until the graduated line 22.5 on scale 7th coincides with the indicating-edge of button B. Read the product 33.3 opposite the I line of one scale Afrom the other one. and mark it by index t' on scale 2d. Draw slide c until the graduated line 1.5 on scale 8th coincides with the indicatingedge of button a. Read the product 93.06 from either scale 7th or 8th opposite the I line of the other one. Add both products by drawing slide c until the graduated line 93.06 on scale 3d coincides with the graduated line 33.30, as indicated on scale 2d by index z'. The snm 126.36, read from scale 3d opposite the 100 line of scale 2d, is the volume (in cubic yards) of one ruiming foot of the prismatoid. The volume of the whole prismatoid, being 100 in length, would be 126.36 cubic yards.

rIhe whole series of operations here described may, with some practice, easily be done in some few minutes, and in shorter time yet, if many calculations have to be carried out in which the same values for n and B enter; or if the values to be operated upon are not too complicated 5 or if the crosssections are of triangular shape, (ditches j' c',) in which case, B

H Jf- L a l is dispensed with; or if the slope a is l, (common cuts,) in which case the multiplicabeing 07 the multiplication BX tion n is spared, or if il disappears.

Ff B z B' sion of the equation Hh n n n Ff for a given numerical value of 7,-, and for B successive values of 7, the average height,

Hh, is expressed by the absciss distance to any point of curve the ordinate of which is For constructing the curves, some numer- B ical values oi 27-, are set out as ordinates by arbitrary scale; and as abscisses are set out by the scale 2d or 3d, those values of Hh which result by substitution in the equa-tion above Ff of those values, 7 for which the curve has to be drawn, the several points of curve set out in this way are connected by a continual line.

F The values for which succeeding curves are constructed, increase by simple arithinetical progression. For the estimator here de- F' scribed the increase of 7g is irom curve to Ff curve: one unit for values between 0 and l0; tive units between 10 and 100; ten units between 100 and 1000; twenty units from 1000,

tween two succeeding values of may be,

, F intervals between two values expressed by succeeding curves are wider on the drawing than stated in the description.

For common practice it is sufficient if the curves are constructed for values 27, O 20, and for values Hh 0 50; but for the rest the curve-diagram may be extended or abridged at pleasure, the exactiiess ot the estimator and the easiiiess of its -practice greatly increasing if the ordinates are set out by as large a scale as possible. I have taken this scale four times greater than the width of slide would have allowed it' the undivided diagram had been iixed on it; and then I have cut the diagram in four ribbons parallelly to the axis of abscisses, which ribbons are iixed on both sides of slide a, the iirst ribbon containing that portion of the-ciirves which com prises the ordinates 0 5; the sec- B B ondZ-n: 5. ..105 thethiri 27,: 10. ..105

the fourth;L l5 20 5 but I wish my patent to cover also estimators with undivided diagrams of curves.

The ciphered graduated lines or marks of degrees alongthe axis of ordinates, as represented on the drawing, express numerical values of 721%; but lately I have constructed estimators where tlieciphers along the axis of ordinates indicate numerical values of while the respective graduated lines are constructed for values B il. Hereby the division ais avoideih bg.

ing operated upon directly.

Vhen making' use ofthe diagram oi' curves the slide c is drawn to the lett until the edge of index d coincides with the zero-line of scale 2d, then the slide ci is put in such a position as that the portion of diagram ot' curves which L contains the numerical value of E entering into calculation stands close before the plate d on slide c.

Along the plate d moves the tongue f, guided by a slot, at any point of which it may be iixed by the brake-screw c. The curves are almost normally transversed by the tongues oblique indicating edge, which is provided with three index-lines. The slide a is in its right position if the most convenient one ot those index-lines meets the axis ot' ordinates of the diagram of curves. Then the tongue f should be moved up or down, and fixed so that the chosen index-line points out the graduated line on the axis oi' ordinates by which the nu- L merical value 7enterin g the calcula-tion is represented. If, now, slide a is kept in position, while slide c isv drawn to the right until the chosen index-line on toiigne f meets that F curve by which the value of j entering the calculation is expressed, then the value ot' Hh, depending on the respective values of F and g, may be read from scale 2d by index d.

, F It any value of Tj has to be operated upon tonguef. Letfi jf be should be drawn until the index-line on ton gue 175, the slide a curve 170.

f points just in the middle between the curves 170 and 180. The space between those curves,

until the index-line on tongue f points 1gdegree before curve 180, or 1% degree behind .E may The equationHi z [if i. 2

B F be written (H102 (Hh) %-f,or, generally,

x2 -l-Y a a0 b.

Of course, the estimator may be used directly for the solution of all second-degree equations of last-named shape, as far as the numerical values a (to be sought along the axis of ordinates) and b (expressed by the curves) are inside the limits of the ciirve-diagram on slide a; ft a 0. 40, and b 0 2500 4500, if the estimator is .of ordinary size.

F f B The divisions ,i haviiig been performed with or without the help of the multiplication (division) scales 7th and 3th, that part of the curve-diagram is placed before the index-plate d on slide c which contains the ordinate value B ,-L entering calculation. Previously, the slide c has been drawn to the left until the edge of index-plate l falls in with the Zero-point of scale 2d, -and with the edge of plate g. The

tongue f is moved up or down until the index-line on same points out that graduated line on axis of ordinates by which numerical B value of enteringcalculation is expressed.

While slide a is kept in this position, slide c is drawn riglitward until the index-line on tongue f meets that curve, or that point between two ciirves, which represents the value The value of hbelonging to the respect- B ive values,-Z and may then be read from F point between two curves, which represents Now, the value H-l-h could be read from scale 2d by index d. But this reading can be (EL-702 spared, because you will nd, directly,

on scale 4th byindex i; T08- on scale 5th H by index g; on scale 6th by index g.

These values, and those of u, B, L, are operated upon exactly as described above for tlie case that H and li were given directly instead of F and f. Given: F=4500Uf5 f=1500lIl 5 :3:22.55 71:15.

From the given values follows, directly,

Draw slide c until the zero-line of scale 2d is touched by the indicating-edge of d. Place right before slide c that part of slide a which B contains the diagram for ordinate values 7,

f meets the curve 3f: 1000; then h 25 may be read from scale 2d by index d, and kept in mind. Slide c being held in place, slide a is drawn rightward until the axis of ordinates is touched by the index-line on tongue f,- then slide a is held in place, while slide c is drawn rightward until the index-liiie on tongue f F touches the curve*7L 3000. From scale 2d could now (by index d) be read 70 78', this being H h. AThe value li 95', as previously read from scale 2d, now having been marked on scale 1st by index t, the following readings are done: From scale 6th, by index g, 1.35 is marked on scale 8th by index B; from scale 4th, by index t', 1.60, and from scale 5th, by index g, 49.05. The sum of both, or 50.65, is marked on scale 8th by button a'.

The multiplication B 1.35 is done by drawing slide c until the 22.5 line of scale 7th is opposite the indicating-edge of button B. The product 30.37, as read from either scale 7th or 8th, opposite the I line of the other one, is marked on scale 2d byindex i'. The multiplication a 50.65 is done by drawing slide c until the 1.5 line of scale 7th is in opposition with the indicating-edge of button a. The product 75.97 is read from either scale 7th or Sth, opposite the I line of the other one. This product is added to the first one by drawing slide c until the 75.97 degree of scale 3d coincides with the indicating-edge of index z'. The sum read from scale 3d opposite the 100 line of scale 2d, viz., 100+6.34=106.34, expresses the volume of each running foot of the prismatoid. i

If the second-degree equation x2+36-lw= 4205 has to be solved, the slide c should be drawn until the edge of index d coincides with the zero-line of scale 2d., Then that piece of curve-diagram showing the ordinates 30 i0 is placed before slide c, so that the index-line on tongue f meets the axis of ordinates, and the tongue is ixed so that its index-line points exactly between the ordinate degrees 36 and 30.2. Now the slide c is drawn (c meanwhile being held in place) until the index-line on tongue f is one-fourth ofthe distance between the curves 4200 and 4220 behind the curve 4200. From scale 2d may now be read, by index fi, fc 40.3.

Invertedly, the estimator may be used for dedncing mechanically from a given volume the average height of the prismatoid containing this volume. Hereby the estimator proves very useful for determining how much the grade of a preliminary railroad-line ought to be attached, or how much such a line ought to be thrown to the side for balancing as much as possible the quantities in the cuts and einbanliments ot'a given railroad-section, provided the ground on the sides ot' the preliminary line previously has been crosssectioned. rlhe different scales may be turned so as to progress in a direction opposite the present one 5 or the scale 1st may be applied on slide c, and synchronously the scales 4th, 5th, 0th on rule b. Then the index-plate g is let't out, the scales 4th, 5th, 0th being read by index d, and the index li is made to move in a groove on slide c; or the scale 1st maybe lei't out, the respective readings then to be done from scale 2d, which, for that purpose, should be furnished with a second ciphering, identical with that on scale lst. Or the scales 4th and 5th may be combined, the spaces between equivalent graduated lines being thrice as wide on scale 4th as they are on scale 5th. 0r the whole instrument may be constructed in the shape of a prism or a cylinder, with the scales parallel to its axis, or in the shape of a disk divided into degrees on its plan or edge, &c.

The drawing represents, in Figure 1, a plan view, and in Fig. 2 a head view, otthe estimator, Fig 3, a longitudinal, and Fig. a a transversal, section oi' same 5 Fig. 5, aview of slide a from both sides.

Z1, rule, in which, by double grooves, move the upper slide c and the lower side a, one above the other, d, index-plate attached to upper slide c; f, tongue sliding along slit ot' index-plate d.; c, brake-screw to iix that tongue with 5 1'., index sliding in groove L along one E, lA, index-buttons slidb 4 5 ingin groove k along other edge of slide c; g, index-plate attached to rule c; 7i., button for drawing slide c. (May be placed elsewhere on the slide c.)

The scales lst to Sth on rule b and slide c are on drawing, but indicated as mentioned in the ingress; likewise the diagram of curves on slide u.

Sums between 50 and 100 are found on this estimator by adding 50 to the value, as read from scale 3d, opposite the 50 line of scale 2d. The multiplication-scales rth and 8th have such an extent as that factors from 0.07 to l0 may by them be operated upon directly, and the diagram ot' curves on the lower slide edge of slide c;

B 1 c 1s constructed for values w 0 40, Il:

625 0 GO, and corresponding ones ot H L=0 25. The graduated ordivisionlines along' the axis of ordinates of the diagram of curves on lower slide of estimator are ciphered by values of "Ii, as referred to in the description, while the corresponding degrees on drawings are ciphered by values of l.

FREDRIC MAURICE STAPFF.

lVitnesses:

NEU. A. EFWING, EUGNE Fonswav.

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