Rare Variant of Genaille's Digital Calculator Click on any image for a larger view.

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GENAILLE'S RODS -- A PERFECTED FORM OF CALCULATING RODS FOR MULTIPLICATION AND DIVISION, French, 1885, signed on the card box "Les Réglettes Multiplicatrices, Appareils à Calculs Exacts et Instantanés pour Simplifier la Multiplication et la Division. Inventés par Henri Genaille et Perfectionnés par Edouard Lucas," published in Paris by E. Belin. The ten wood rods, each 6-3/4" (17 cm) long, are covered on all four sides with printed paper columns, and are used in conjunction with the fixed rod. The results are instantaneous, for numbers as large as ten digits. The set is complete and in excellent condition, the original box with instructions, fine.

These rather complex rods each present, using the four sides, tables and diagrams for four different digits (identified on the top and bottom of each rod). There are nine rows, aligning with the numerals 2 through 9 on the left-most (fixed) rod. Black triangles align with graded columns of digits to give multiplication tables, etc. In use, for multiplication one simply selects the rods to form any multiplicand, starts with the right-most digit, and follows the path of the triangles to read out the result.

The inventors developed various aides for calculation, all apparently published in 1885. We have had their "Réglettes Néperiennes" (Tesseract Catalogue 72, Item 41), their "Financières" rods for commercial calculations (36 /47), their "Multisectrices" for division (47/51), and now for the first time their stunning set of "Multiplicatrices." An extremely rare, ingenious variant. (10322) $3950.


The Origin of Digital Calculating Click on any image for a larger view.

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ORIGINAL SET OF NAPIER'S CALCULATING RODS, English, late 17th / early 18th century, of variously toned boxwood throughout, contained in the original 4-1/8" x 2-5/8" x 5/8" (10.4 x 6.5 x 1.7 cm) case with lovely incised geometric decoration to both sides. This fine all-original set comprises 16 four-sided calculating rods, the tabulat carrier, wood case and slide-off lid. All the numerals are punched, and in an interesting idiosyncrasy the maker had mis-labeled four of the "4" multiplication table columns as "8", and then corrected them. The numeral punches all match, and the set is absolutely complete and all original, and in very fine condition. This set was made with a full 16 rods (for multiplying very large numbers), instead of supplying a square/cube rod. Inscribed in ink in the tabulat is a tally of the number of sides available (either 6 or 7) for each digit from 0 to 9, for the total of 64 sides.

In 1617 John Napier (1550 - 1617), of Merchiston, Scotland, published Rabdologiae, in which he revealed his newly invented calculating rods capable of rapid multiplication of very large numbers. Each "bone" is finely punched with a multiplication table for the integer on top multiplied by 1, 2, ...9. Diagonal lines show how to carry the second and further digits. The four sides of each bone present tables for four different integers. Thus one can multiply numbers up to 16 digits long with this set! (10331) $14,500..


Rare Swiss Mathematical Sector, c.1750 Click on any image for a larger view.

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EIGHTEENTH CENTURY SWISS SECTOR, c. 1750, signed "I.C. Bartenschlager, Schaffhavsen." This substantial mathematical sector is all brass, opening fully to 14" (36 cm). It is hand-divided and labeled (with all punched numerals and letters) on both sides with a full set of computational scales, including the doubled sector scales of sines, tangents, secants, cubes, chords, metals, polygons, arithmetic and geometric scales, plus edge scales of sines, tangents, logarithmic numbers, cannon ball diameters and cannon bore diameters. Condition is very fine, with traces of gilding. Bartenschlager is recorded in the Websters' Index for a ring dial, a table dial, a sector, and surveying instruments. Zinner records his dates (1713 -1799), and his production of mathematical instruments. Schaffhausen is first mentioned in 1045, and developed into a fine Renaissance city north of Zurich in northernmost Switzerland. We have here a rare example of an early Swiss instrument . (9332) $4500.


The "Numerical Pocket Piece who teaches abroad, or privately at home," Click on any image for a larger view.

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THE "ARITHMETICAL MEDAL AND NUMERICAL POCKET PIECE," English, 1753, "Sold by I * Maddux, at the hand & pen in Brook Street, Holbourn." The cast bronze medal, 1-3/4" (45 mm) in diameter, carries on one face the "Numerical Pocket Piece who teaches abroad, or privately at home," with its "Table for Multipl'n. & Division by a new & short method," giving all integral products from 3 x 3 up to 9 x 9. The other face includes a table converting pence to shillings and pence, a table of multiples of twelve, and a "numeration table" giving place value of units, tens, ..., hundred millions. Condition is very fine, the medal retaining much of its gilding. A most unusual arithmetical teaching aide mémoire / pocket calculator. (7344) $1150.


By an 18th century Delft Instrument Maker Click on any image for a larger view.

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EARLY DUTCH MATHEMATICAL RULE, second half 18th century, signed "I. Reghter, Delft." This thick brass rule, 14-3/4" (37.5 cm) long, is divided on both sides with a total of six strongly nonlinear scales plus one scale of (Delft?) inches running from 0 to 12, the inches divided respectively into halves, thirds, fourths,..., twelfths (each inch 26.3 mm long). Scales are labeled cryptically "S,N" (1-32, 1-100), "V,R" (1-32, 1-125), and "L,T" (1-32, 1-125). Condition is fine except for several stains to the rule. We do find Reghter's trail in the references -- Zinner lists him as "I. Rechter a Delft," known for a planetarium (in the "Leiden Museum", #A182). King and Millburn describe this as a double-cone planetarium, and refer to Jan or Johannes Reghter (1730 - 1801) of Delft, "a highly skilled maker of compound microscopes, geometrical instruments, and electrical apparatus." Reghter was a published scientist in his own right. He designed and built an atmospheric electrometer which won a prize contest in 1786, and which led to a quarrel when van Breda, who had commissioned Reghter to develop this instrument, took all the credit and the prize (see Zuidervaart, 2006). We have had one other instrument by this maker, an unusual mechanical level with integral scale rule also with the undecoded "V,S,L" scales (Tesseract Catalogue 76, Item 34). (8309) $2950.


A Diversity of Slide Rules I Click on any image for a larger view.

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RADIAC SLIDE RULE, FOR NUCLEAR FISSION PRODUCTS, English, mid-20th century, 11-3/4" (30 cm) white plastic with two double-sided slides plus cursor, excellent and fascinating. $595.

MINIATURE, probably American, 2nd quarter 20th century, stamped "Buckeye Lamp Division, Cleveland, Ohio," 4-1/2" (11 cm) long plastic over wood, double-sided slide plus cursor, logarithmic and trigonometric scales plus ruler, fine. $120.

RICHMOND'S IMPROVED MILK SCALE, English, c. 1915, signed "Daisy Supply Co., Ltd., Park Royal, London, N.W. 10," 12-1/2" (32 cm) long, brass-bound boxwood, scales of Lactodensimeter Reading, Specific Gravity, Temp., Fat, and Total Solids, fine with one chip, rare. $395.

COMBINED SLIDE RULE AND RULERS, Japanese, 1961, so dated and signed "Scientific Instruments Co.," Model 1050, 11-5/8" (30 cm) long, plastic laminate over bamboo, double slide, cursor, table to reverse, original leather case, excellent. $275.

DICAS HYDROMETRY RULE, English, c. 1815, signed "Dicas Patent Improv¹d by Gill & Co. London," 7-3/8" (19 cm), ivory, slider and scales both sides, Water, Alcohol, Proof, etc., very fine with slide warping. Invented by John Dicas in 1780. SOLD

UNUSUAL SOLID IVORY VINTNER¹S SLIDE RULE, English, c. 1815, signed "J.&G. Quin, Patentees," 10" (25 cm), complex with scales both sides including "Sikes¹s Concentrated Strengths," temperature corrections, etc., excellent. (10393) SOLD


A Diversity of Slide Rules II Click on any image for a larger view.

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"KEL-CO CLASS CALCULATOR FOR THOROUGHBRED RACE HORSES," American, c. 1975, 10" (25 cm) plastic, with case and instruction booklet, fine but cursor stuck, very complex computations for the winning wager! $295.

"CHRONOPOSE EXACT DE CANNEVEL," French, c. early 20th c., 7-1/2" (19 cm), paper over wood with brass, one slide for exposure calculations, the other for actual exposing of photographic test strips, good with wear. $170.

THE "UNIQUE" LOG-LOG SLIDE RULE, English, c. 2nd quarter 20th c., 6-3/8" (16 cm), celluloid over mahogany, slide and cursor, trig tables to reverse, good. $150.

COMPLEX CIRCULAR SLIDE RULE, American, c. 1940, 4-1/8" (10 cm) diameter, printed on enameled aluminum both sides, with three rotating cursors, excellent. $250.

CLASSIC PLASTIC SLIDE RULE, American, c. 1950, copyright by Ecco Co., a diminutive 6-5/8" (17 cm) long, with twin-sided cursor and scales. $95.

UNCOMMON KEUFFEL & ESSER SLIDE RULE, American, c. 1920, 10-5/8" (27 cm), plastic laminate to mahogany, table to reverse, beveled linear rule, double-sided slide, cursor with unusual readouts atop and aside for the three logarithmic scales of singles, squares, and cubes, the latter along the side of the rule, very fine with good leather case. $350.

THE POST "VERSALOG" RULE, Japanese manufacture, "Hemmi," model 1460 by The Frederick Post Co., 12-1/2" (32 cm), plastic laminate to bamboo, highly complex, double sided, counting 23 scales, with cursors and original leather case, excellent. $330.

EARLY FOUR-SLIDE RULE, English, c. 1800, 12" (30 cm), lovely hand stamped boxwood throughout, four sided with four sliders, the latter each with scales or constants or ratios on the inner surfaces, all to enable calculation of numerous mathematical functions and the volume content of partially filled vessels of various shapes, very fine. (10473) $595.


Print Your Own -- a Rare Production Plate Click on any image for a larger view.

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BRASS PRODUCTION PLATE FOR SLIDE RULES, American, 1961, signed in the plate "Keuffel & Esser Co., K-12 Prep, K&E 4158." The substantial 12-1/4" x 2-3/4" (31 x 7 cm) brass plate bears in relief the complete highly detailed layout for this single-sided slide rule, with all the divisions, calibrations, and labels. There are nine scales, including those on the slider which here is formed integrally in its zero position on the rule. Condition is excellent. This most rare factory production plate was used for the model 4158 (K12 Prep), which was released in 1961. It was supplanted by the model "68-1892 (K12 Prep)" the following year; consequently it was marketed and sold, but apparently never illustrated in the K&E catalogues. Rare. (10483) $1750.


Art and Mathematics in the 17th century Click on any image for a larger view.

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EXCEPTIONAL MATHEMATICAL SECTOR, European, probably German, c. late 17th century. Opening to 17-1/4" (44 cm) this substantial brass sector is constructed with an unusual "sandwich" hinge of concentric disks. The inner disk is hand engraved with a wonderful "rosace" on both sides. The head of each arm is engraved with sinuous floral designs, different on the two sides. There are three different doubled sector scales on each side, including one pair with the planetary symbols of metals. Condition is generally fine noting some nicks and scratches.

The decoration on this sector is some of the most appealing we have seen, but identifying the country of origin is difficult. We see considerable similarities to early German sectors, yet the floral images are similar to those on a Blondeau French pearl sizer, and on those on an early Edmund Culpeper English set we had (Tesseract Catalogue G Item 55). A pleasing mystery. (10493) $9500.


Sixteen Local Measures Compared, finely subdivided on a Long Scroll Click on any image for a larger view.

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REMARKABLE LINEAR MEASURE COMPARISON SCROLL, German, early 19th century, signed "Leipzig, im Magazin für Industrie und Literatur." A heavy card tube, 9" (23 cm) long, is bound in red paper (simulating Morocco leather), bearing labels for the "Ellenmaass = Tabelle" within, plus identifications of sixteen different linear measures for the ell, mostly local German ones. Included are the Augsburg Elle, the Brabant Elle, the St. Gallen Leinwand, the Russian Archine, and even the French Meter. Rolled within the tube, and wound up by wood handles, is a 42-1/2" (108 cm) long thick paper scroll with sixteen very detailed scales which align with the legends on the tube. The scales run variously from zero up to anywhere from 35 to 64 units, with subdivisions every tenth as well as every twelfth unit. Condition is fine noting some exterior soiling and some small paper losses on the tube ends.

According to the labels, this unusual scroll gives comparisons of sixteen of the most common ell measures used in commerce, both German and foreign. The ell was originally defined as the length of the forearm from elbow to tip of middle finger. It evolved into many different regional standard measures, typically about a yard long.

A most rare device, the only example we have encountered. (9298) $4500.


With Fine 17th c. Decor and Letter Forms Click on any image for a larger view.

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EARLY FRENCH SECTOR, c. second half 17th century, the substantial brass rule opening on its three-leaf sector hinge to 13-3/8" (34 cm) overall. The hinge is finely engraved with a six-lobed rose on each side, and the near corners of the sector likewise have fine leaf decoration. There are few scales -- only two doubled sector scales per side -- as often the case with the earliest sectors. Exquisitely labeled and with hand-punched numerals, we find "Lignes des Partis Esgalles, Les Plans, Lignes des Cordes, Les Sollides." In fine condition, noting a few stains and with a fine old patina, this is a rare example of early French mathematical instrument. (9309) $2200.


Substantial, Precision Logarithmic Slide Rule Click on any image for a larger view.

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UNCOMMON ALL-BRASS SLIDE RULE, French, mid-19th century, unsigned but probably by Richer, stamped on the case "Règle Logarithmique." Measuring a full 17" (43 cm) overall, this substantial rule is made of solid brass, apparently, with a heavy non-tarnishing gray finish. It has a reversible slider, all precision divided with logarithmic "Nomb." scale, and with transcendental trigonometric functions "Sinus," "Tang," and "Caract" scales. Condition is very fine, complete with the original leather covered wood case, the ends notched for use of the calculating rule in place. (9278) $1500.


English, Spanish, French Measure Compared Click on any image for a larger view.

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UNUSUAL COMPARISON RULE, English, c. second quarter 19th century, signed "Troughton & Simms, London." This 13-1/2" (34 cm) long brass rule is 5/16" square in cross section, engraved on the four sides with scales of English inches from 0 to 12 by tenths, Spanish inches likewise, old French inches (the old "Pouces du Roy") from 0 to 12 by twelfths, and "Decimetre's" from 0 to 3 by hundreths (i.e., by millimeters). Condition is excellent, complete with its original slightly domed mahogany case.

The partnership of Edward Troughton and William Simms commenced in 1826, and was a long-lived manufactory of all manner of optical and mathematical instruments. We have had a number of interesting comparison rules, but none quite like this. It may have assisted in English trade with France and Spain, and was made at a time when the old French inch measure was not yet fully abandoned in favor of the metric system (which was not made mandatory until 1840 -- see item 31 in this catalogue). (9279) $975.


Measuring the Partial Contents of Tapered Vessels Click on any image for a larger view.

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PORTABLE IRON GAUGER'S ROD, c. early 19th century. Opening on two hinges from 18-5/8" to 50-1/2" (47 to 128 cm), the rod has a well-shaped crook-neck handle or belt hanger on one end, and tapered point on the other. One side bears a calibrated linear scale divided every centimeter from 0 to 100, the other a very nonlinear scale again divided every unit from 0 to 100. Here the divisions get closer and closer together proceeding from the bottom; the separation between 0 & 1 is 257 mm, 1 & 2 is 66 mm, and finally 99 & 100 is 4 mm. Condition is fine noting one clip broken.

The rod was probably designed to measure the quantity of liquid in a partially filled vessel of a specific shape. The nonlinear scale suggests volume measurement in a strongly tapered vessel with very narrow base, something like an amphora, and is the first of its kind we have encountered. An interesting project would be to measure all separations, assume a circular cross section, and plot the profile of the intended vessel. (10321) (ON HOLD)


Boxwood by Brander, c. 1770 Click on any image for a larger view.

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UNUSUAL SECTOR BY BRANDER, German, c. third quarter 18th century, finely signed on the brass sector hinge "G.F. Brander." This large sector opens to a full 25-3/4" (65 cm), finely made with arms of boxwood, three leaf sector hinge of brass and steel, and brass terminals on the ends. The arms bear twin sector scales of equal parts, divided every unit from 0 to 200, punch-numbered every 10 units. Condition is excellent except for considerable warping of the arms.

Georg Friedrich Brander (1713 - 1783) was probably the finest and most prolific German scientific instrument maker of his time, constructing a wide variety of innovative astronomical instruments, microscopes, surveying, navigation, and demonstration instruments, etc., and publishing extensively on the subject (see Brachner et al., 1983). No single-purpose sector like the present one is evident in the extraordinary collections described by Brachner, but in a famous painting Brander seems to be holding just such an instrument (Brachner p.23).

The sector¹s double ³line of lines,² as it is often called, is particularly useful for subdividing, using a pair of dividers, any given line into any number of equal parts, or in fact to solve for proportional parts. A good description is found in Hutton¹s A Philosophical and Mathematical Dictionary (1815). (10302) $2400.


Incorporating a Full Traverse Table for Plane Sailing at Sea, c. 1700 Click on any image for a larger view.

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EARLY FOUR FOLD WANTAGE RULE WITH NAVIGATING / CALCULATING SCALES, English, c. 1700, signed "Peter Brock" within fleur-de-lys punches. The rule opens, with its three inset brass three-leaf hinges, to 36" (91 cm). There are external scales for inches (0 to 36 by tenths), for "Butt. Lying" (from 0 to 108, presumably for cask wantage), for "Beer Gallons" (0 to 125) and "Wine Gallons" (0 to 102). One "internal" side has a long nonlinear scale running 5 to 36, and a reversed nonlinear scale of 9 to 36, with short terminal arrays of (1, 2, 3, 4, 5, 6, 7), (_, 36, 16, 9, 5, 4, 2), (1, 2, 3, 4) and (12, 6, 4, 3). The final (internal) side has tables of (I, F) and (F, (I), presumably feet and inches, another of pounds, shillings and pence (from 1p 2s 4d to 6p 57s 0d), and a well designated double table for calculating the "Difference of Latitude" (i.e., the distance in miles between the parallels of latitude of any two places), and the "Departure" (the distance in miles between the meridians of two places, measured at constant latitude). Each measure is given here for every quarter point in four points of the 32-point sailing rose, i.e., for every 128th of a circle, or every 2.8125 degrees. Thus one has a full traverse table for plane sailing, as published in greater detail in most early books of navigation (see e.g., Robertson, The Elements of Navigation, 4th ed., 1780). Condition of this rule is worn, darkened, and a bit warped, although it is undamaged and the scales are essentially all quite readable. A significant rule, by this newly found maker. (7314) $3950.


The Metric System Compromised Click on any image for a larger view.

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ENORMOUS METER-LONG MATHEMATICAL SECTOR, French, c. 1820, signed "Compas (Viret)", and "No. 9." This major instrument measures 4-1/8" x 20-3/4" x 5/32" (10 x 53 x 0.4 cm) closed, and opens to a full 39-7/16" (100 cm). It has a three-leaf sector hinge, and closes on the central steel alignment leaf and the three alignment pins. One side is divided with a linear scale from 0 to 12 by twelfths, labeled "Pied Ancien" representing the 18th century standardized "Pied de Roy" (27.1 mm / inch), plus a double sector scale of equal parts divided from 0 to 57 by ones. The reverse has (1) a similar but unlabeled "Foot" scale of 12 inches by twelfths of an inch (at about 26.8 mm / inch), (2) a linear "Pied Métrique" scale also running from 0 to 12 by twelfths (and curiously at 27.9 mm / inch), and (3) a double sector scale of equal parts divided from 0 to 220 by ones. Condition is fine, noting numerous small nicks and scratches. This remarkable rule is housed in its softwood case with slide off lid.

The "Pied Métrique" scale on this sector represents a fascinating period in the transition to the metric system in France. An inch of 27.9 mm gives a foot of approximately 335 mm (actually intended to be 333-1/3 mm), and thus a meter of exactly 3 feet. This was part of the "mesures usuelles" introduced by Napoleon in 1812, and finally abolished in 1839. The formal metric system had been gradually introduced into France, département by département (Paris being first), between 1795 and 1800. It was poorly understood by the populace, unpopular, and little used in local trade. As a compromise, Napoleon introduced this sort of "metric" system based on the old measures. A meter thus equaled one yard of three traditional feet, the liter was subdivided as a quart, the kilogram was specified as exactly two pounds, etc. It was a confusing journey into the eventual acceptance of the greatly simplified true metric system. The engraved word "Viret" in parentheses is something of a puzzle, but probably indicates the maker of this grand sector. A clockmaker by this name is indeed listed in Paris directories during the 1820's, and a splendid gilt clock by him is known. (9366) $4950.


Rare Multifunction Slide Rule for Chemistry Click on any image for a larger view.

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CHEMICAL ENGINEERING SLIDE RULE, Japanese, c. 1960, signed "Hemmi, Japan, No. 257." The 12-1/2" (32 cm) long rule is made of laminate over bamboo, metal bound, and fitted with double-sided sliding glass cursor. The two sides of rule and slider are divided with a total of 23 scales, including gauge marks for determination of atomic and molecular weights, temperature, conversion scales, pressure unit conversions, molar / weight / volume fractions, adiabatic compression and expansion, etc. In addition there are the standard scales of multiplication, roots, logarithms, exponents, etc. The outfit is in near new condition, complete with instruction manual and leather case. The Hemmi Slide Rule Company was established in 1895 (first as Henmi Jirou & Co.) They became a major manufacturer of both standard and specialized slide rules, introducing the model 257 in 1954. But within 10 years the introduction of electronic calculators initiated the precipitous decline in slide rule manufacture, and the Hemmi company eventually turned to electronics. (7316) $950.


A Most Rare Patented Calculator Click on any image for a larger view.

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THE "CALCULATEUR DIDELIN," French, c. last quarter 19th century, "Brevete France S.G.D.G. et Etranger." The 19-1/2" x 8-1/2" x 2-1/2" (50 x 22 x 6 cm) softwood case, with hinged lid and sides, contains a system of five lithographed tin rollers, plus a hinged overlay plate with a total of 150 (!) windows opened and closed by 30 independent shutter slides. The rotating tubes (for units, tens, hundreds, thousands, and ten-thousands) are printed with interest due for 30 different percentage rates (from 0.25% up to 9%). Condition is fine throughout noting a little wear and soiling. It is an intriguing and quite rare specialized calculator; we find no example in the published catalogues of the collection in the Science Museum in London, and but one in the Musee des Arts et Metiers, that one given to the Museum by Monsieur Didelin himself in 1892. (7337) $3500.


Patented French Calculator Click on any image for a larger view.

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THE "TACHYLEMME," French, c. 1880, signed "C.L. Chambon, Inventeur; Lith. Baster & Vieillemard, 97, B't. Port-Royal, Paris; Approuvé par la Societe d'encouragement pour l'industrie Nationale." This patented calculator has a fine 4" x 6-1/2" (10 x 16.5 cm) polished wood body with ebonized wood base, beveled glass top, and plated metal frame. The lithographed panel displays, through windows, the printed numbers on four cylindrical rollers for 1's, 10's, 100's, and 1000's. There are windows for percentage rates (of interest, etc.) of 1% to 6% by halves. External knobs allow setting any four digit number; one then sums the four displayed values for the result. Condition is fine noting a slight chip to the glass. Baster and Vieillemard are well known as fin-de-siècle Parisian lithographers of color postcards, publicity cards, etc. Chambon himself invented the "Tachylemme" in 1876. An uncommon device -- one is recorded as having entered the IBM collection. (9365) $1450.


Musical Scale on an Early Sector Click on any image for a larger view.

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A MUSICAL SECTOR, German or Italian, c. late 17th century, made of brass, reverse tapered in thickness, opening from 7-1/2" to 14-1/8" (19 to 36 cm). One side is finely divided with twin sector scales of equal parts (each scale divided twice 0(1)100; i.e., from 0 to 100 by ones), of "Lin Recta" 15(1)2, of "Lin 'square'" 0(1)60(5)90, and of a wonderful musical scale "fi, G, G-is, a, b, h, C, C-is, D, D-is, E, f." The other has twin sector scales of "Lin Cub" 1(1)50(5)120 and "Lin Circul" 60(5)45(1)6, noting a couple of tiny terminal fleur-de-lis marks . Condition is good with general surface wear.

This interesting sector has standard scales for computing distances, areas, etc. But it also has a musical scale of 12 notes, spaced nonlinearly and representing the differences in pitch. It is specifically German nomenclature, distinguished by the "-is" abbreviation sign (engraved here somewhat like the Greek beta) meaning "sharp", the "b" for B-flat, and the zigzag-h for B-natural (Jeffrey Dean, 2012; and see W. Apel, 1949, The Notation of Polyphonic Music, 900-1600). Such a musical scale can have many uses in the design and construction of instruments, for example calculation of lengths of lute strings or organ pipes, of string gauges or tensions, or even of bell diameters. The only other musical sector scales we have found are on very large German sectors (e.g. see R. de Pecker, 2007, and a particularly fine example at Oxford). (9395) $4500.


"Arithmetical Not Zoological" Click on any image for a larger view.

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FIRST FORM OF WEBB'S ADDER, American, c.1875, signed "C.H. Webb, N.Y., 'The Adder,' Patd. March 10th, 1868," #C671. Made of sheet brass on mahogany, 6-3/4" (17 cm) overall, this is the uncommonly found early form of Webb's "two-wheeled velocipede of figures." Described in detail by Kidwell in Rittenhouse (1, 12), it was succeeded by Webb's popular all-metal version. This is a good example of this American calculating instrument, in fine condition, with photocopies of Webb's instructions and glorious testimonials. The New York Times of 2 Jan. 1869 headlines "'The Adder' -- Arithmetical Not Zoological." And from Inventors and Manufacturers'Gazette of the same month, "This 'Adder' can easily be worked in a room where a dozen persons are engaged in conversation, and the operator himself can talk during the adding process without fear of blunders." Webb himself seems to have been a man of many talents. Under the pen name "John Paul" he was the "wise and witty New York correspondent" to the Springfield, Massachusetts Republican; he authored Wickedest Woman in New York, and was the playwright for numerous burlesque performances. (9303) $975.


Specialized "Napier" Rods from The Netherlands Click on any image for a larger view.

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SPECIALIZED CALCULATING RODS -- "NAPIER'S BONES" FOR SIZE AND WEIGHT CONVERSIONS, Dutch, c. 1800, with the partial trade label of "Dirk Van Den Bosch," mathematical instrument maker of Rotterdam. The fine mahogany case contains this complete set of 20 four-sided, 8-3/8" (21 cm) long, boxwood rods. Each rod is numbered on top with an integer from 0 to 9, and punched on each face with four sets of columns of nine numbers, each column headed by different lettered abbreviations. The latter are identified on a label in the case, being RV (Rheinland foot), AV (Amsterdam foot), HE (The Hague ell), HTP (Holland troy pound), AP (Amsterdam pound), NE (Netherlands ell), and NP (Netherlands pound). Further abbreviations have these letters preceded by V (for squares) or K (for cubes). In addition to the printed trade label and table of ratios (Verhoudingen) in the box, there is an old manuscript explanation written (in Dutch) by an early user. This note refers to van Swinden, the famous Dutch mathematician and physicist who wrote a major work on weights and measures, and was instrumental in bringing the metric system to the Netherlands. Condition is excellent noting some rods warped, and an old crack in the case lid.

This is the first such set we have encountered. In using it we find, for example, that one side of the "1" rod has columns for RV, NE, HE, and NE. The first column gives, to nine decimal places, the number of Rheinland feet in one Netherlands ell; the second column the reciprocal; the third the number of The Hague ells in one Netherlands ell; the fourth the reciprocal of that. The same face on the "2" rod has the same entries doubled, and so on to the "9" rod. Assembling a group of rods, and adding along the diagonals as with Napier's rods, we find directly how many of one unit are in any very large number of the other unit of length, or weight. With the "V" and "K" scales we can do this direct conversion for areas and for volumes, respectively.

The maker / retailer of this ingenious set is not found in the standard references. (9352) $9500.


Immediate Volumetric Computation Click on any image for a larger view.

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SHEPPARD'S PATENTED DOUBLE-SLIDE SLIDE RULE, English, c. 1880, signed for the maker "Stanley Great Turnstile Holborn London," for the inventor "Fred'k. A. Sheppard Patentee No. 160," and for the owner "Claude W. Atkinson." Constructed of fine boxwood, 13" (33 cm) overall, the rule has two two-sided slides running alongside each other. In the simple mode for volumetric calculations, the upper fixed scale "Length," the two sliding scales ("Breadth" and "Thickness"), and the lower fixed scale "Cubic Content" all have identical logarithmic divisions. To find the volume of a rectangular parcel, for example, one sets up the three dimensions and sees the answer immediately. To the reverse of the sliders are scales of square roots and linear scales of numbers. One edge of the rule has an applied (later?) paper scale of sines. We note a discussion of the relative merits of Sheppard's rule versus one of Dring and Fage, published in the 1883 English Mechanic and World of Science. An uncommon rule, in very fine condition throughout. (9342) $1150.


Calculator in Your Pocket, c. 1900 Click on any image for a larger view.

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THE CALCULIMETRE CHARPENTIER, French, c.1900, signed "Calculimetre, G. Charpentier, Brevete S.G.D.G." The 2-3/8" (6 cm) diameter calculator is made of nickel silver and brass, divided with logarithmic, square root, tangent and sine scales on both sides, the suspension affixed to twin cursors and a frontal circular scale. A milled edge allows one to rotate the main body against this suspension assembly. Condition is fine, the surface somewhat cleaned and scratched. The Charpentier calculator, a fine pocket slide rule, is analogous to the Mannheim design but with circular scales. It was covered by a French patent, and by a British one in 1882, and was listed in American Keuffel & Esser catalogues from 1895 up to 1927, and in Dietzgen catalogues (see Otnes, 1991). Despite this, surviving examples are relatively uncommon; here we are able to offer a good French one, complete with photocopies of instructions and of the original French patent papers. (9331) $1250.


Complex Early Calculating Sector Click on any image for a larger view.

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MICHAEL SCHEFFELTS' MATHEMATICAL SECTOR, German, early 18th century, unsigned. This substantial brass sector is 7" (18 cm) long (closed), opening on a three-leaf hinge engraved both sides with a fine rose design. The faces bear a prolific number of single scales and doubled sector scales, for all manner of practical calculations in applied mathematics and engineering, including gunnery use, fortification design, surveying, navigation, plane and solid geometry, densities of materials, etc., etc. The numerals are punched in the brass, but the scale labels, the metallic symbols, and all the vignettes of polyhedral solids are boldly hand engraved. Condition is very fine noting light scratches to the surface.

The choice of scales, labels, and layout design on this sector is essentially identical to that illustrated in Michael Scheffelts' comprehensive book Instrumentum Proportionum..., published in Ulm, Germany, in 1708 (see illustration above). The few very minor differences show that the sector maker understood the mathematics, e.g., tangent scale running 0° to 45° (rather than 0° - 65° in Scheffelts), avoidance of engraver's error in the book (on the Cubic scale), etc. Michael Scheffelts was born in Ulm in 1652, and worked in several positions including teacher of mathematics. He was a very fine craftsman, and is known for several good instruments (see Zinner and Webster's Index). His book on the sector explains, in 148 pp and 12 full page plates, the construction and practical uses of his many scales. A rare example of Scheffelts' sector. (9311) $8500.


From the Thirty Year's War Click on any image for a larger view.

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SEVENTEENTH CENTURY GUNNERY GAUGE, German or Austrian, 1629, boldly engraved "Anno 1629." Rectangular in cross section, and tapered, the brass rule measures 13-1/2" (34 cm) long overall, including its cast and soldered double-sided bust handle. Each face of the rule is hand engraved, one with a scale of inches (labeled "Zol") from 0 to 12, each inch subdivided into the corresponding integral number of parts (e.g., the eleventh inch is divided into 11 equal parts!) Each inch measures approximately 24.1 mm, and is thus consistent with the local foot measure in a number of German city states (see Gilliland, SIS Bulletin 20, for an overview). The other three sides are divided nonlinearly, giving the weight of a cannonball of the measured bore diameter, for "Eisen Kugel" (Iron Balls) from 0 to 125 "Pfunt" (pounds), for lead balls (also over the range 0 to 125), and for stone balls (0 to 60 pounds). Condition is very fine.

Related gauges, one even with a bust, exist in the Dresden princely collections, illustrated in Grotzsch & Karpinski (1978, fig. 140). The present one dates from the Thirty Year's War (1618 -1648), which raged throughout Europe and devastated Germany. It is intriguing to seek an identification of the bust, assuming it is specific, but so far this is too speculative. What we do have is a wonderful dated survival from the early 17th century. (8377) $19,500.


Complex Scales by Engineer to Brunel Click on any image for a larger view.

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METFORD'S QUADRUPLE ENGINEERS SCALES IN IVORY, English, late 19th century, signed "Metfords Engineers Pocket Scales" and "Hudson & Kearns, London." The four ivory rules are each 6-1/2" (17 cm) long, of triangular cross section, and divided on all three sides with scales, constants, measured values, equations, etc. There are simple linear charting scales very finely divided (e.g., "6 inches to a mile"); relationships in triangles ("Hypoth= Base x Sec Angle at Base"); "Ordnance Scale;" expansion measurements for mercury, for glass tubes, for steel rods, etc; specific gravity of palladium, aluminum, etc; the logarithm of the force of gravity; the one-hundredth of a second of arc at the equator, measured in English feet; etc. We count 17 divided scales and numerous useful facts. Cryptic abbreviations on the ends of each rule form a table of contents. Condition is excellent except for a small chip at one end of one rule. The original leather bound case is present, lacking its end piece.

The inventor was presumably William Ellis Metford (1824 - 1899), a dedicated engineer who worked under Brunel for a time, and who is known for his improved surveying theodolite (see his high quality miniature theodolite "The Metford Theodolite with Repeating Circle," Tesseract Catalogue 64 Item 23.) One example of "Metford's Double Set of Ivory Pocket Scales suitable for Civil Engineers, Architects, and Land Surveyors" is presumably in the Science Museum -- we find it listed in the Catalogue of the Educational Division of the South Kensington Museum (1867). That set was signed by Pastorelli & Co.; ours by Hudson & Kearns, known especially as publishers in late Victorian England. A rare example of this complex set. (8357) $3950.


A Splendid Calculating Rule in Silver Click on any image for a larger view.

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EXQUISITE SILVER SECTOR, Continental, possibly Italian, c. early 18th century, bearing a punch mark, and with lovely shaped ends, and roses engraved to the hinge. Overall length is 4" (10 cm) closed, and the sector is beautifully engraved with doubled sector scales "Arithmetica, Geometrica, Cubica, Gradus Quadrantis." Additional scales give "demy Pied Roy," i.e., half the French linear measure of the King's Foot with subdivisions in inches and twelfths; 'Reinlendscher. 1/2 Fusz," i.e., half the Rhineland foot, likewise subdivided; and reminders of the sizes of one-pound cannon balls of iron, and of lead. The words are all beautifully hand engraved with flourishes, and the tiny numerals are all hand punched. Interestingly, the scales and their names are all read with the hinge to the right, rather than to the left as on typical French and English designs. Condition is very fine.

This is a diminutive example of an elegant form occasionally seen in brass; the design, execution, and terminal shaping may be compared with a Dutch sector by Metz (Tesseract Catalogue 46 Item 45) and with a c. 1700 Continental sector (Catalogue 71 Item 36). The only example we have seen in silver. (8267) $4800.


Complex Slide Rule for Cattle, Oxen, Sheep, Swine Click on any image for a larger view.

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EWART'S ANIMAL GAUGE, English, c. 1840, signed "The Cattle Gauge & Key to the Weighing Machine, arranged by John Ewart, Newcastle upon Tyne; J. Tree Maker, 22 Charlotte St., Blackfr¹s. Rd., London." Measuring 9" (23 cm) overall, this double sided rule is made of brass-bound boxwood with ivory slider. The numerous tables and scales allow one to calculate the yield of meat from various types of cattle, oxen, sheep, and swine, depending on their live weight, length, girth, class of ox or bull, degree of fat, whether shorn, etc. This very finely made complex rule is in fine condition noting a bit warped with age. (7326) $950.


A Complex Circular Calculator Click on any image for a larger view.

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LORD'S CALCULATOR, English, c. 1900, signed "R. Waddington, Coventry." The 2-3/4" (7 cm) plated brass watch case is glazed in front, and fitted with two winding knobs and side mounting block. The calculator has four concentric circular scales (labeled A, B, C, and D), divided on silvered brass, the numerals colored in red and black. Two of these scales are independently rotatable by the knobs; no cursor is used. Condition is very fine and functional, noting a little wear to the finish. Lord's sturdy pocket calculator is quite rare -- even Hopp records only its existence, without details. (7299) $1350.


The Power of Radioactivity Click on any image for a larger view.

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RADIATION SLIDE RULE, French, mid-20th century, signed "SCRGR; CEA; GMP." The 13-3/4" (35 cm) long plastic rule is double faced, with double sided slider and cursor. It is divided with scales and nomographs for source activity, irradiation, intensity (over 10 powers of 10), photon energy, source distance, material (iron, concrete, aluminum, water, etc.) In fine condition, with the original leatherette case. (7349) $395. (ON HOLD)




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