Calculation 
Early Circular Calculating Rule  Click on any image for a larger view. Scroll to view more items. 
EARLY CIRCULAR SLIDE RULE, English, first quarter 19th century, signed "Fraser, London." The 4" (10 cm) lacquered brass disk houses an inner silvered brass disk with inset wire thumb grip for rotation, and with large clamp screw. The outer disk is engraved with a circular 110 logarithmic scale (labeled "G"); the inner with a similar scale (labeled "L"). The reverse is beautifully engraved with the owner's emblem of a stag head and an entwined pair of initials "S.F." Condition is very fine and functional, noting a slight bend and slight binding to the main disk. The original wooden case, covered in red Morocco leather and lined in green velvet, is fair, with warping and detachment of the hinge. The Fraser family (Alexander, H., J., William I, William II) were active instrument makers in London over the period 1780 to 1828 (see Clifton), listed consistently at addresses on Wardour Street in Soho, and at number 3 New Bond Street. Linear slide rules were produced in significant quantities from the mid 17th century until the mid 20th. The earliest circular and spiral logarithmic slide rules do date from the 17th century, but relatively few were made even in the 18th and 19th centuries. They gained popularity in the late 19th century, especially for their compactness, and in the 20th century as many forms of special purpose calculating rules. The present rule is a rather rare early 19th century example. (9356) (SOLD) 
Incorporating a Full Traverse Table for Plane Sailing at Sea, c. 1700  Click on any image for a larger view. Scroll to view more items. 
EARLY FOUR FOLD WANTAGE RULE WITH NAVIGATING / CALCULATING SCALES, English, c. 1700, signed "Peter Brock" within fleurdelys punches. The rule opens, with its three inset brass threeleaf 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 32point 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. Scroll to view more items. 
ENORMOUS METERLONG MATHEMATICAL SECTOR, French, c. 1820,
signed "Compas (Viret)", and "No. 9." This major instrument
measures 41/8" x 203/4" x 5/32" (10 x 53 x 0.4 cm) closed,
and opens to a full 397/16" (100 cm). It has a threeleaf 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 3331/3 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. 
The "Numerical Pocket Piece who teaches abroad, or privately at home,"  Click on any image for a larger view. Scroll to view more items. 
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, 13/4" (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. 
Rare Multifunction Slide Rule for Chemistry  Click on any image for a larger view. Scroll to view more items. 
CHEMICAL ENGINEERING SLIDE RULE, Japanese, c. 1960, signed "Hemmi, Japan, No. 257." The 121/2" (32 cm) long rule is made of laminate over bamboo, metal bound, and fitted with doublesided 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. Scroll to view more items. 
THE "CALCULATEUR DIDELIN," French, c. last quarter 19th century, "Brevete France S.G.D.G. et Etranger." The 191/2" x 81/2" x 21/2" (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 tenthousands) 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. 
The Combining Power of Compounds  Click on any image for a larger view. Scroll to view more items. 

THE FIRST SLIDE RULE OF CHEMICAL EQUIVALENTS, English, c. 1815  1820, signed "Publish'd by W. Cary, 182 Strand, Jan. 1, 1814." The rule is made of thick printed paper applied to mahogany and varnished, measuring 21/2" x 125/8" x 3/16" (6 x 32 x 0.5 cm) overall. The slider scale is logarithmic, from 10 to 320. The body has two lists of chemical compounds (92 in all), each compound placed in the position of its quantitative combining power, based on oxygen at 10. The reverse has applied a very early owner's handwritten instructions by example, and the rule is complete with its early sewn chamois carrying sleeve with belt(?) loop (a remarkably early forerunner of the mid20th century nerd's slide rule belt case). This is all contained in a modern protective mahogany box. Condition of the rule is good, noting water stains and slight edge wear to the paper. This is a rare example of William Hyde Wollaston's 1814 slide rule, enabling straightforward calculations to find what weights of different elements or molecules combine to produce a given product weight (or conversely how much of each material results from decomposition). Wollaston presented his "Synoptic Scale of Chemical Equivalents" to the Royal Society in 1813, and William Cary (the important instrument maker) published his rule two months later. We have had two other early paper on wood chemical slide rules, one developed by Scholz in Vienna in 1821 (Tesseract Catalogue 68 Item 36), the other by Ehrmann in the 1820's (Catalogue 85, Item 36). But this is the first example of Wollaston's original we have been able to offer. More details on chemical slide rules are given by W. Williams (1992), who was able to locate only six other Wollaston ones surviving. (9385) (SOLD) 
Patented French Calculator  Click on any image for a larger view. Scroll to view more items. 
Musical Scale on an Early Sector  Click on any image for a larger view. Scroll to view more items. 
A MUSICAL SECTOR, German or Italian, c. late 17th century, made of brass, reverse tapered in thickness, opening from 71/2" to 141/8" (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, Gis, a, b, h, C, Cis, D, Dis, 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 fleurdelis 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 Bflat, and the zigzagh for Bnatural (Jeffrey Dean, 2012; and see W. Apel, 1949, The Notation of Polyphonic Music, 9001600). 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. Scroll to view more items. 
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, 63/4" (17 cm) overall, this is the uncommonly found early form of Webb's "twowheeled velocipede of figures." Described in detail by Kidwell in Rittenhouse (1, 12), it was succeeded by Webb's popular allmetal 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. Scroll to view more items. 
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 foursided, 83/8" (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. Scroll to view more items. 
SHEPPARD'S PATENTED DOUBLESLIDE 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 twosided 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. Scroll to view more items. 
THE CALCULIMETRE CHARPENTIER, French, c.1900, signed "Calculimetre, G. Charpentier, Brevete S.G.D.G." The 23/8" (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. 
By an 18th century Delft Instrument Maker  Click on any image for a larger view. Scroll to view more items. 
EARLY DUTCH MATHEMATICAL RULE, second half 18th century, signed "I. Reghter, Delft." This thick brass rule, 143/4" (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" (132, 1100), "V,R" (132, 1125), and "L,T" (132, 1125). 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 doublecone 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. 
Complex Early Calculating Sector  Click on any image for a larger view. Scroll to view more items. 
MICHAEL SCHEFFELTS' MATHEMATICAL SECTOR, German, early 18th century, unsigned. This substantial brass sector is 7" (18 cm) long (closed), opening on a threeleaf 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. Scroll to view more items. 
SEVENTEENTH CENTURY GUNNERY GAUGE, German or Austrian, 1629, boldly engraved "Anno 1629." Rectangular in cross section, and tapered, the brass rule measures 131/2" (34 cm) long overall, including its cast and soldered doublesided 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. Scroll to view more items. 
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 61/2" (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 onehundredth 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. Scroll to view more items. 
Complex Slide Rule for Cattle, Oxen, Sheep, Swine  Click on any image for a larger view. Scroll to view more items. 
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 brassbound 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. Scroll to view more items. 
The Power of Radioactivity  Click on any image for a larger view. Scroll to view more items. 
Rare Swiss Mathematical Sector, c.1750  Click on any image for a larger view. Scroll to view more items. 
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 handdivided 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. 
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