About the Work
Universal Arithmetic is a comprehensive mathematical treatise by Edmond Halley, presented here in a nineteenth-century Glasgow edition published by W. R. M‘Phun. The work is designed as a companion volume to every set of mathematical tables, with particular reference to the Practical Mathematician’s Pocket Guide. The book combines theoretical investigation with practical application, covering logarithms, trigonometry, mensuration, astronomy, and navigation.
The text is structured as a systematic mathematical manual, beginning with an introduction explaining the construction and use of logarithms, followed by sections devoted to the application of logarithmic methods to arithmetic and geometry, the investigation of trigonometric rules, and finally astronomical and nautical problems. The preface, dated 11th December 1838, from Blythswood Hill Mathematical Academy in Glasgow, indicates the author’s pedagogical intent and his desire to provide a work that would serve students and practitioners alike.
The volume includes extensive tables for surveyors, mariners, astronomers, and engineers, retained from the Practical Mathematician’s Pocket Guide to keep that work affordable. The book’s scope is ambitious, encompassing everything from the theoretical foundations of logarithmic systems to practical rules for finding longitude at sea by lunar observations. The author acknowledges taking a “freedom” in his derivation of the word “logarithms” to convey its modern acceptation, hoping to avoid imputation of ignorance of the term’s history.
The edition also contains prefatory material advertising other publications by the same firm, including guides to domestic cookery, commercial book-keeping, merchant’s and banker’s practice, and practical mechanics. These advertisements reveal the context of educational publishing in early nineteenth-century Glasgow and the market for practical handbooks aimed at tradesmen, merchants, and students.
Context
The work emerges from a tradition of practical mathematical education in Britain during the eighteenth and nineteenth centuries. Halley’s treatise, originally composed in an earlier period, was republished in Glasgow in 1839 as part of a series of pocket guides intended for working professionals. The publisher, W. R. M‘Phun, specialized in compact, affordable manuals covering domestic and commercial subjects, as evidenced by the advertisements preceding the main text.
The mathematical content reflects the state of logarithmic and trigonometric knowledge in the period. The work engages with the Neperian system of logarithms, named for John Napier, and discusses the common system based on the number 10. The author demonstrates familiarity with the Differential Calculus and Taylor’s Theorem, using these tools to investigate trigonometric formulae. References to Woodhouse’s Trigonometry indicate engagement with contemporary mathematical literature.
The astronomical sections reflect the practical needs of navigation in the age of sail. The work includes methods for determining longitude at sea through lunar observations, a technique of paramount importance for maritime commerce and naval operations. Examples drawn from the Nautical Almanac for 1835 and 1838 show the currency of the methods presented. The inclusion of observations by Lacaille at the Cape of Good Hope and Wargentin at Stockholm in 1751 demonstrates the international character of astronomical research in the period.
The educational context is significant. The author, Robert Wallace, conducted a mathematical academy at Blythswood Hill in Glasgow, and the work is clearly shaped by pedagogical concerns. The preface indicates that the introduction explaining logarithms was largely reproduced from an earlier work, the Practical Mathematician’s Pocket Guide, with new matter added. This suggests a deliberate effort to create a self-contained textbook that could serve students working without extensive prior mathematical training.
The publishing context is equally important. The advertisements for other M‘Phun publications reveal a market for practical guides aimed at a broad readership. The Pocket Guide to Domestic Cookery went through five editions, the Pocket-Guide to Commercial Book-keeping through three editions of 3000 copies each, and the Merchant’s and Banker’s Commercial Pocket Guide through six editions of 6000 copies. These figures suggest a substantial audience for practical handbooks, and the mathematical guide was clearly intended to appeal to the same market.
Structure of the Book
The work is organized into four main sections, preceded by an introduction and followed by extensive tables. The introduction explains the construction and use of logarithms, including methods of calculating them by common arithmetic. This introductory material is substantial, occupying a significant portion of the early part of the book and establishing the conceptual foundations for everything that follows.
Section I contains mathematical investigations of the rules presented in the introduction, including algebraic formulae for logarithms and trigonometrical functions. The latter are established via the Differential Calculus and Taylor’s Theorem. This section represents the theoretical heart of the work, demonstrating why the rules work rather than merely stating them. The investigations cover the derivation of formulae for the sine and cosine of sums and differences of angles, the factorization of the sine series, and the establishment of logarithmic series expansions.
Section II applies logarithms to common arithmetic, including compound interest and annuities, mensuration of lines, surfaces, and solids, and plane and spherical trigonometry. This section is practical in orientation, providing worked examples of multiplication, division, fractions, proportion, and other operations performed using logarithms. The examples cover a wide range of applications, from calculating the cost of goods to determining the area of fields.
Section III covers investigations of plane and spherical trigonometry rules, mensuration of heights and distances applied to astronomy and surveying, and gives a prominent place to Napier’s Rules for Circular Parts and his Analogies. This section also includes astronomical and nautical problems, including finding longitude at sea by chronometers and lunar observations. A rule is provided for finding field areas using tables of difference of latitude and departure.
Section IV provides tables for surveyors, mariners, astronomers, and engineers. These tables include logarithmic sines, tangents, and secants to every point and quarter point of the compass; lengths of a degree of longitude on parallels of latitude; the sun’s parallax in altitude; the moon’s parallax in altitude corrected for refraction; mean refraction of heavenly bodies; depression or dip of the horizon; dip at various distances; augmentation of the moon’s semidiameter; difference between true and apparent level; and distance of horizon corrected for refraction.
The table of contents details the structure of the work, listing sections on powers and roots, the nature and use of logarithms including the Neperian system, skeleton tables, formulae for calculating logarithms, and investigations of trigonometric formulae. The contents further detail problems in mensuration, including parallelepipeds, cylinders, cones, spheres, spheroids, and casks; plane and spherical triangle solutions; astronomical problems such as sun and moon distances, parallax, and dip of the horizon; and surveying.
Detailed Section-by-Section Summary
Advertisements for M‘Phun’s Pocket Guides
The volume opens not with mathematics but with a series of publisher’s notices for W. R. M‘Phun’s series of pocket guides, of which the featured work is the latest. The first of these is The Pocket Guide to Domestic Cookery, by a Lady, now in its fifth edition at one shilling and sixpence. This edition adds an entire chapter on Preserves and includes instructions for trussing with illustrative plates. Its advertised merit is that it avoids the bulk and expense of larger cookery manuals while supplying everything necessary for a well-ordered kitchen. A review quoted from the Paisley Advertiser praises M‘Phun as an indefatigable guide, recalling his earlier Guides to the Lions of St Mungo, horse purchase, Mothers, health, and Stenography, and declares this Cookery guide “more important than all the rest put together.”
Next is The Pocket-Guide to Commercial Book-keeping, third edition, of which three thousand copies have been printed, by Robert Wallace, A.M., of the Blythswood Hill Mathematical Academy. This treatise covers Merchants’ Accounts by Single and Double Entry, with definitions, the books used in business, Jones’s System, ledgers, balance sheets, and sections devoted to Inland and Foreign Trade. Reviews from the Berwick Warder, the Scottish Guardian, and the Glasgow Argus commend its cheapness, simplicity, clarity, and the uniqueness of its plan. Following this is The Merchant’s and Banker’s Commercial Pocket Guide, sixth edition, six thousand copies, dedicated to Alexander Goodsir, Esq., Secretary to the British Linen Co.’s Bank. This manual treats banking, broking, exchange, insurance, public funds, and commercial transactions, with chapters on mercantile terms, bankruptcy, the banking systems of the Bank of England, Scotland, Ireland, and foreign countries, the East India Company, exchange, prices, fairs, funds, and insurance. The Tyne Mercury, the Belfast Northern Whig, the Glasgow Chronicle, and the Scotsman all praise its small size, cheapness, and condensed utility.
The final advertisement is for The Practical Mechanic’s Pocket Guide, sixth edition, six thousand copies, dedicated to Robert Napier, Esq., of the Vulcan Foundry, also by Robert Wallace. It contains rules and tables for calculating the working effects of prime moving machinery, the strength of materials, and mechanical powers. The Athenaeum particularly values its chapter on water-power, suggesting that steam may not merit exclusive preference, while the Perth Courier recommends it to landed proprietors for hints on disposing of water power.
Preface to The Universal Calculator’s Pocket Guide
The preface, dated from Blythswood Hill Mathematical Academy, Glasgow, 11th December 1838, explains the work’s purpose: it is intended as a “Companion to every set of Mathematical Tables,” especially the “Practical Mathematician’s Pocket Guide.” The introduction from that earlier work’s first edition, which explained logarithms, is largely reproduced here with new matter added. The Introduction explains the construction and use of logarithms, including methods of calculating them by common arithmetic. Section I contains mathematical investigations of the introduction’s rules and algebraic formulae for logarithms and trigonometrical functions, the latter established via the Differential Calculus and Taylor’s Theorem. Section II applies logarithms to common arithmetic, including Compound Interest and Annuities, Mensuration of Lines, Surfaces, and Solids, and Plane and Spherical Trigonometry. Section III covers investigations of Plane and Spherical Trigonometry rules, Mensuration of Heights and Distances applied to Astronomy and Surveying, with a prominent place given to Napier’s Rules for Circular Parts and his Analogies, and astronomical and nautical problems including finding longitude at sea by Chronometers and Lunar Observations. It also contains a rule for finding field areas using tables of Difference of Latitude and Departure. Section IV provides tables for surveyors, mariners, astronomers, and engineers, retained from the “Practical Mathematician’s Pocket Guide” to keep that work cheap. The Author notes he has taken a “freedom” in his derivation of the word “logarithms” to convey its modern acceptation, hoping to avoid imputation of ignorance of the term’s history.
Table of Contents
The table of contents details the structure of the work, listing sections on powers and roots, the nature and use of logarithms including the Neperian System, skeleton tables, formulae for calculating logarithms, and investigations of trigonometric formulae. The contents further detail problems in mensuration—parallelepipeds, cylinders, cones, spheres, spheroids, and casks—plane and spherical triangle solutions, and astronomical problems concerning sun and moon distances, parallax, and dip of the horizon, as well as surveying. A second table of contents lists logarithmic and astronomical tables, designated C through M, with their page numbers. These include logarithmic sines, tangents, and secants to every point and quarter point of the compass; lengths of a degree of Longitude on parallels of Latitude; the Sun’s parallax in altitude; the Moon’s parallax in altitude corrected for refraction; mean refraction of heavenly bodies; depression or dip of the horizon; dip at various distances; augmentation of the Moon’s semidiameter; difference between true and apparent level; and distance of horizon corrected for refraction.
The Nature and Use of Logarithms
The main text begins with an Introduction explaining logarithms. Section 1 states that the term derives from two Greek words, logoi and arithmon, literally meaning indices of numbers; logarithms are defined as a system of artificial numbers indicating or representing natural numbers. Section 2 claims that logarithms replace tedious arithmetic: multiplication and division become addition and subtraction; involution and evolution—raising powers and extracting roots—become simple multiplications and divisions.
Sections 3 through 14 explain the nature of powers and roots. A number multiplied by itself gives its square or second power; the number itself is the root or first power. Multiplying the second power by the root gives the cube or third power, and so on; this process is involution. Indices indicate how often the root enters as a factor. If a number is considered a power of another, the latter is called such a root; finding roots is evolution. The reciprocal of a number is a fraction with numerator unity and denominator the number itself. The reciprocals of indices are indices of roots—for example, the square root index is ½, the cube root index ⅓—so roots are fractional powers. Using 2 as root, successive powers are calculated by multiplication: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024. Powers are denoted by placing indices on the right of the root, as 2¹=2, 2²=4, and so on. Roots are denoted similarly, as 4^½=2, 8^⅓=2. Multiplication of powers or roots of the same number is indicated by adding indices; division by subtracting indices; involution by multiplying indices; evolution by multiplying indices by required root indices or dividing by denominators. If any power or root divided by itself gives unity, the index of unity is 0; hence 2⁰=1, unity being the zero power.
Sections 15 through 23 describe the nature and use of logarithms. Powers of any assumed invariable root constitute natural numbers; their indices are artificial numbers or logarithms, taken collectively a system; the assumed root is the base. The logarithm of a given number is the index of that power of the base equal to the given number. Any number may be base; with base 2, a table of logarithms is constructed: 1 has logarithm 0, 2 has 1, 4 has 2, 8 has 3, 16 has 4, 32 has 5, 64 has 6, 128 has 7, 256 has 8, 512 has 9, 1024 has 10. Examples follow: to multiply 16 by 8, add logarithms 4 and 3, sum 7 gives product 128; to multiply 4, 8, and 16, add 2, 3, 4, sum 9 gives 512. To divide 256 by 64, subtract 6 from 8, remainder 2 gives quotient 4. To find a fourth proportional to 8, 32, 64, add logarithms 5 and 6, subtract 3, remainder 8 gives 256. To find the square of 16, multiply logarithm 4 by 2, product 8 gives 256. To find the cube root of 64, divide logarithm 6 by 3, quotient 2 gives root 4.
The Common System of Logarithms
Sections 24 through 42 cover the common system. The number 10 is assumed as base because it is the root of the decimal scale. Powers of 10 are unity with ciphers annexed: 10⁰=1, 10¹=10, 10²=100, and so on. A skeleton table lists natural numbers 1, 10, 100, 1000, with logarithms 0, 1, 2, 3. Dividing unity by successive natural numbers gives decimal fractions .1, .01, .001; their logarithms are found by subtracting from 0, indicated as negative: -1, -2, -3. These are negative logarithms; the logarithm of every proper fraction is essentially negative, increasing numerically as the fraction decreases; the logarithm of 0 is negative infinity. Extracting successive square roots of 10 gives roots 3.16228 (index ½), 1.77828 (¼), 1.33352 (⅛), 1.15478 (1/16), forming a Table of Even Roots. Extracting successive cube roots gives 2.15443 (⅓), 1.29155 (1/9), 1.08902 (1/27), 1.02883 (1/81), forming a Table of Odd Roots. A Third Skeleton Table has Part I from eighteen successive square-root extractions and Part II from eleven cube-root extractions, listing natural numbers with decimal logarithms. By these tables, logarithms of all natural numbers may be found. To find the logarithm of any prime number, Rule 1 divides the prime by the nearest tabular number less, continuing until the quotient coincides with a tabular number; the sum of logarithms of the divisors and final quotient gives the required logarithm.
Decimal Indices and the Neperian System
Articles 43 through 46 discuss decimal indices. A table of decimal place-names—Tenths, Hundredths, Thousandths—is presented, followed by a table showing how the index of a logarithm changes as a number’s value is lowered in the decimal scale. Numbers from 100100 down to .00001001 are paired with logarithms whose decimal parts remain constant at .000434 while the index falls from 5 to 7, with bar notation used for negative indices. The common system’s advantage lies in its base being the same as the root of the decimal scale: increasing or diminishing the index by unity immediately gives the logarithm of a decimal multiple or submultiple. Thus, tabulating decimal parts for integers from 1 to 10,000 or 100,000 suffices for innumerable numbers, with indices supplied by rule. Any other base would lack these advantages, requiring separate calculation and tabulation of logarithms with indices for fractions, integers, and powers of the base; the reverse operation—finding numbers from logarithms—would be even more difficult. Tables give average differences of every five logarithms in an adjoining column, enabling approximation of logarithms for numbers beyond the table, based on the principle that differences of nearly equal numbers are nearly proportional to differences of their logarithms. The example from Part I, art. 31, is cited: numbers 1.00056, 1.00028, 1.00014 have differences .00028 and .00014, with logarithm differences .000122 and .000061; the proportion .00028:.00014 :: .000122:.000061 is correct as far as the decimals extend, though only nearly correct if extended further—sufficiently accurate for practice.
Articles 47 through 59 address the Neperian system. Napier’s system, sometimes improperly called Hyperbolic, is theoretically the most natural, with base 2.718281828459, expressible only by the infinite series 1 + 1/1 + 1/(1·2) + 1/(1·2·3) + 1/(1·2·3·4) + &c., otherwise written 2 + 1/2 + 1/6 + 1/24 + 1/120 + &c. Logarithm construction depends on the Logarithmic Series, where a number’s logarithm in every system is the same infinite series united to a factor called the Modulus—a constant function of the base. If an integer is assumed as base, the modulus becomes an infinite series, as in the common system; if an integer is assumed as modulus, the base becomes an infinite series, as in the Neperian system. The Neperian system’s peculiarity is its modulus being unity, rendering logarithms independent of the base—hence “most natural”; Napier could construct his logarithms without reference to any assumed base, a remarkable discovery made half a century before the Logarithmic Series was known. Logarithms of the same number in different systems are proportional to their moduli; since the Neperian modulus is unity, the common modulus is found by proportion: Nep. log 10 : com. log 10 :: mod. Nep. : mod. com.; as common log 10 = 1, the fourth term is 1/(Nep. log 10)—the modulus of the common system is the reciprocal of the Neperian logarithm of its base.
Article 52 introduces a verbal rule, a translation of a useful form of the Logarithmic Series, for constructing either Neperian or common logarithms, universally applicable and converging more rapidly as the given number increases. Article 53 gives the full rule: to find the Neperian logarithm of a number, given the Neperian logarithm of the difference between that number and unity, divide unity by the difference between double the given number and unity for a first quotient; divide this quotient by the square of that difference for a second quotient, and so on; divide these quotients respectively by the odd numbers 1, 3, 5, 7, 9, 11, 13; sum as many quotients as have significant figures two or three decimal places beyond the required accuracy; double this sum, add the Neperian logarithm of the difference between the given number and unity; the result is the Neperian logarithm. Multiply by the common modulus to get the common logarithm.
Article 54 works Example 1: the Neperian logarithm of 2. Subtract unity from 4 (double 2), divide unity by 3, giving quotients .33333333, .03703704, .00411523, .00045725, .00005081, .00000565, .00000063, each divided by 9 successively; dividing these by 1, 3, 5, 7, 9, 11, 13 gives .33333333, .01234568, .00082305, .00006532, .00000565, .00000051, .00000005; sum .34657359; doubled = .69314718; adding the Neperian logarithm of 1, which is 0, leaves .69314718 as the Neperian logarithm of 2. Article 55, Example 2, finds the Neperian logarithm of 4, the square of 2, as 2 × .69314718 = 1.386294.
Series for π and Sine Factorization
Articles 89 through 93 present series for π and sine factorization. The passage opens mid-calculation, stating that 4x — A = .7854 and a- = 3.1416, the latter being the circumference of a circle with diameter unity, the former also its area. These results come from taking only three terms of the value of 4x and one of A; taking eight terms of the former and three of the latter yields π = 3.141592653589793.
Article 90 presents the sine series: sin x = x(1 — x²/2·3 + x⁴/2·3·4·5 — &c.). Setting sin x = 0, x may be 0, π, 2π, 3π, or —π, —2π, —3π. Treating this series as an equation, these values are its roots. Substituting —π²/y and reducing so the highest-dimension term, y⁷ for instance, stands first, the roots become —π²/π², —π²/2π², —π²/3π², and —π²/—π², —π²/—2π². Consequently y — 1, y — ½, and y + 1, y + ½, are divisors. Separating into factors and dividing by y, we obtain sin x = x(1 — x²/π²)(1 — x²/4π²)(1 — x²/9π²) &c.
Article 91 puts x = mπ/2n, giving a factored expression. Taking logarithms of the factors yields: log sin(m/2n · 90°) = log m — 3 log n — 3 log 2 + log(2n + m) — log(2n — m) + log(1 — m²/6n²) + &c. Article 92 cites art. 68, formula V, giving a logarithmic series expansion. Article 93 states that summing the coefficients of m², m⁴, &c. obtains the first formula in art. 60, the second obtained similarly, referencing Woodhouse’s Trigonometry, p. 337 et seq.
Application of the Tables: Logarithmic Arithmetic
Section II applies the tables to common arithmetic. Article 94 covers Multiplication: add the logarithms of the factors; the sum is the logarithm of the product. With negative indices, the difference between their sum and the positive indices’ sum is the index. Units carried from decimal parts are always positive. Examples: 125 × 256 = 32000; the continuous product of 1.25, .0016, and 18.25 = .0365. Article 95 covers Division: subtract the divisor’s logarithm from the dividend’s; the remainder is the quotient’s logarithm. Rules for handling indices of like or different signs are given. Examples: 96000 ÷ 768 = 125; 75 ÷ 400 = .1875; .15625 ÷ .625 = .25. Article 96 treats Fractions: subtract the denominator’s logarithm from the numerator’s to find a vulgar fraction’s logarithm and decimal equivalent. Example: 651/1736 = .375. Article 97 treats Mixed numbers: reduce to an improper fraction, or find the decimal equivalent of the fraction and annex it to the integer. Example: 9⅞ = 9.875. Article 98 treats Parts of an integer: from the logarithm of the given number of parts, reduced to lowest denomination, subtract the logarithm of the number of such parts in the integer. Example: 17 sh. 10½d., or 858 farthings, = .89375.
Article 99 covers Practice: add the logarithm of the price to the logarithm of the quantity. Example: 486 yards at 18 sh. 7½d. costs 452L. 11s. 9d. Article 100: subtract the quantity’s logarithm from the cost’s. Example: 37 cwt. 2 qrs. costing 131L. 5s. gives 1s. 1¼d. per lb. Article 101: subtract the price’s logarithm from the cost’s. Example: 10L. 6s. 3d. at 5s. 6d. per yard buys 37½ yards. Article 102 covers Proportion: from the sum of the second and third terms’ logarithms subtract the first term’
Plane Triangles: Rules and Examples
Rule 2 (Art. 171): To find a side of a plane triangle when a side and two angles are given, the text presents a logarithmic rule. The rule states that one must add the logarithm of the given side, the arithmetical complement of the logarithmic sine (or the cosecant minus 10) of the angle opposite the given side, and the logarithmic sine of the angle adjacent to the given side. The sum, minus 10, is the logarithm of the side opposite the adjacent angle. The notation is established: angles are denoted by capital letters A, B, C, and the sides opposite them by the small letters a, b, c. An example is worked out for a triangle ABC where a = 104 yards, b = 120 yards, and the angle B = 67° 23′. The computations yield the angle A = 53° 8′, the angle C = 59° 29′, and the side c = 112.
Case II (Art. 172): This case addresses the problem of finding the remaining parts of a triangle when two sides and the included angle are given. Rule 1 is used to find the other two angles. It employs the logarithmic tangent of half the sum of the unknown opposite angles, the arithmetical complement of the logarithm of the sum of the two given sides, and the logarithm of their difference. The sum of the unknown angles is first found by subtracting the given included angle from 180°. Rule 2 then finds the third side, using the logarithmic cosine of half the sum of the unknown angles, the arithmetical complement of the logarithmic cosine of half their difference, and the logarithm of the sum of the given sides. An example is given with a = 290, b = 410, and the included angle C = 105°. The solution yields the angle A = 30°, the angle B = 44°, and the third side c = 560.14.
Case III (Art. 173): When the three sides of a triangle are given, three rules are presented for finding the angles. These rules use the logarithms of the differences between the sides and the semiperimeter to find the half-angles, employing the sine, cosine, or tangent functions. An example is provided with sides a = 679, b = 537, and c = 429. The computed angles are A = 39° 10′, B = 52° 15′, and C = 88° 35′. As a proof of the calculation’s accuracy, these three angles sum to exactly 180°. Art. 174 confirms that these rules are sufficient for solving all plane triangles. A final example of a right-angled triangle at C, with hypotenuse c = 960 and angle B = 56° 15′, yields the angle A = 33° 45′, and the sides a = 533.34 and b = 798.21.
Spherical Triangles
Napier’s Rules (Art. 175): The text introduces Napier’s Rules for solving right-angled spherical triangles. The sine of the mean part is equal to the sum of the tangents of the adjacent parts, or the sum of the cosines of the remote parts, minus 10. The circular parts are defined as the two legs, the complement of the hypotenuse, and the complements of the two oblique angles. An example is given for a right-angled spherical triangle at C, where the angle A = 75° 36′ and the side a = 64° 3′. The results are the angle B = 34° 39′, the side b = 31° 51′, and the hypotenuse c = 68° 11′.
Oblique Spherical Triangles (Arts. 176-177): For an oblique spherical triangle where three sides are given, Art. 176 instructs the use of the log sines of the differences between the semiperimeter and the sides. An example with sides a = 16° 24′, b = 161° 36′, and c = 157° 22′ yields the angles A = 47° 10′, B = 136° 22′, and C = 88° 34′. Conversely, Art. 177 addresses the case where the three angles are given, and the sides are to be found using the log cosines of the half-sum and the differences. An example with angles A = 56° 52′, B = 72° 13′, and C = 107° 47′ produces the sides a = 60°, b = 80°, and c = 100°.
Napier’s Analogies (Arts. 178-182): Six rules, known as Napier’s Analogies, are presented for solving oblique spherical triangles. These rules relate the tangents of half-sums and half-differences of angles and sides, and the sines of sides and angles. Two examples are worked out. In the first, given sides a = 59° 13′, b = 81° 17′, and the included angle C = 125° 36′, the results are the angles B = 62° 9′, A = 50° 13′, and the side c = 114° 38′. In the second example, given angles A = 66° 40′, B = 41° 32′, and the included side c = 60° 51′, the results are the sides a = 53° 18′.85, b = 35° 23′.15, and the angle C = 90°.
Investigation of the Rules (Section III)
Art. 183: The text begins a formal investigation of the rules just presented. The fundamental proposition is that the sides of a plane triangle are proportional to the sines of their opposite angles. This is demonstrated by circumscribing a circle around the triangle ABC, drawing radii to the angles, and constructing perpendiculars to the sides. Arts. 185-191 derive the formulae for the sine and cosine of the sum and difference of angles. Formula I is sin(A ± B) = sin A cos B ± cos A sin B, and Formula II is cos(A + B) = cos A cos B ∓ sin A sin B. From these, formulae III through XIV are derived. Formula XV leads to a key analogy: the sum of two sides is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their difference. This analogy founds Rule 1 of Art. 172. Arts. 192-194 derive formulae XVI through XXII, which lead to another analogy: the cosine of half the difference of two angles is to the cosine of half their sum as the sum of the opposite sides is to the third side. This founds Rule 2 of Art. 172. Arts. 195-199 provide the demonstration of the rules in Art. 173, beginning with Formula XXIII, which states a² = b² + …
Horizontal Parallax
Art. 207: The text defines horizontal parallax using a figure where C is the Earth’s centre, H is a heavenly body on the horizon of a spectator at E, and the angle EHC is the horizontal parallax. With H denoting this angle, D the distance CH, and T the Earth’s radius CE, the relation R : sin H :: D : T gives D sin H = T (setting R = 1), hence D = T/sin H. In logarithmic terms, log D = log T + 10 − log sin H. Two examples are given. The first calculates the Moon’s mean distance from its mean horizontal parallax of 57′ and the Earth’s mean diameter of 7912 miles, yielding a mean distance of 238,603 miles. The second calculates the Sun’s mean distance from its mean horizontal parallax of 8″.5776, yielding a mean distance of 95,129,500 miles.
Parallax in Altitude (Art. 208): For a spectator at E, with zenith Z, a body at P, and the horizon EH, the difference between the zenith distances as measured from the surface and from the centre is the parallax in altitude, represented by the angle EPC. This angle is always less than the horizontal parallax EHC and vanishes at the zenith. From triangle ECP, the relation sin P : sin Z :: T : D gives sin P = sin H sin Z. Since the sines of small arcs are nearly proportional to the arcs themselves, the parallax in altitude P is approximately equal to H sin Z, meaning it equals the horizontal parallax multiplied by the sine of the zenith distance, or equivalently, the cosine of the altitude.
Determining Horizontal Parallax (Art. 209): The method for determining horizontal parallax involves two observers at E and B under the same meridian, with Z and Z’ their respective zeniths. The angle ECB is the difference in latitude. In the quadrilateral ECBP, the angle EPB, denoted Q, is found by the formula Q = Z + Z’ − (L ± L’), where the sign depends on whether the latitudes are of different or the same names. With P = H sin Z and P’ = H sin Z’, the horizontal parallax is found by H = Q/(sin Z + sin Z’). An example is given using observations of the Moon’s north limb made by Lacaille at the Cape of Good Hope and Wargentin at Stockholm on 5th November 1751. Their zenith distances were 56° 19′ 40″ and 38° 4′ 52″ respectively, and their latitudes were 33° 56′ 3″ S and 59° 20′ 31″ N. This calculation yields a horizontal parallax of 60′ 35½”, which varies between the extremes of 61′ 24″ and 53′ 48″.
Error Analysis (Art. 210): Using Taylor’s Theorem on the formula H = D tan E, the text analyzes how a small error e in the angle E produces an error h in the computed height H. The error is minimized when sin 2E = 1, which occurs when E = 45°, meaning the height equals the distance. At an elevation of 30°, the height is half the distance, and at 60°, it is double. An example shows that a 1-minute error at an angle of 45° produces only a 1719th part error in the computed height.
True vs. Apparent Level
Arts. 211–212: The text distinguishes between the true and apparent level. In a figure, the arc DB is the true level, the line DA is the apparent level, and AB is their difference. With T = 3960 miles as the Earth’s radius, A the height, and d the distance, the relation is 2TA + A² = d². Omitting the negligible A² term, the height A is found to be ⅔d² feet for d in miles. Tables are given for the differences in inches for distances from 1 to 9 miles (8, 32, 72, 128, 200, 288, 392, 512, 648) and in fathoms for leagues (1, 4, 9, 16, 25, 36, 49, 64, 81). Conversely, the distance is found by d = √(1.5A).
Dip of Horizon (Arts. 213–214): The line MN, parallel to the tangent at B, is the true horizon, while AD is the observed horizon from a point A. The angle MAD, the Dip, equals the angle ACD at the Earth’s centre. The tangent of the Dip is d/T. The rule for finding the distance to the horizon is to add half the height to itself and extract the square root for the distance in miles. An example is given: a deck 24 feet above the sea gives a distance of 6 miles and a Dip of 5′ 12″. After deducting ⅓ for refraction, this matches the nautical table value. The text notes that refraction depends on air temperature and quotes Professor Woodhouse’s remark that exact formulae are “tools finer than the work.” Simms observes that tabular dips differ from truth, especially in the tropics, so navigators use the Dip Sector. An example shows an observed dip of 4′ 41½” versus a calculated 5′ 12″, a difference of 31″ due to refraction, which is about ⅒ of the true dip. Refraction is noted to vary from ⅛ to ⅕ of the observed dip.
Distance Between Inaccessible Objects
Arts. 215–217: The text presents a method for finding the distance between two inaccessible objects, C and D, from two stations, A and B, in the same plane. The base AB is measured, along with the angles BAC, BAD, ABD, and ABC. Four calculations yield the distance CD, with Formulas I–IV giving logarithmic solutions. An example with base AB = 2436 yards and given angles yields CD = 13293 yards. Alternative Formulas V–VI use the tangent and secant of an auxiliary angle T. A second example involves two ships sailing from the same port, one N.E. by N. for 84 miles and the other S.E. for 76 miles, with an angle of 9 points (101° 45′) between their courses. The distance between them is calculated to be 123.8 miles. The text advises using Formulas III–IV when the sides are nearly equal and the angle is large, and Formulas V–VI when the angle is small.
Cyclic Quadrilateral (Arts. 218–222): For a quadrilateral ABDC that can be inscribed in a circle, with sides a, b, c, and d, the text provides a formula for the cosine of angle A.
Distance Between Celestial Bodies
The text presents a trigonometric formula for finding the distance between two celestial bodies: tan 6 = cot d cos (a — a’) and cos D = sin d sin (d’) sec 6. It notes that when declinations are south, and when a is less than a’, the arcs d, d’, and (a — a’) must be considered negative. An example is given from the Nautical Almanac for 1838, for 1st January at mean noon. The sun’s right ascension is 18 hours, 46 min., 36.02 sec., and its declination is 23° 1′ 49″.4 S.; the moon’s right ascension is 23 hours, 18 min., 4.64 sec., and its declination is 7° 20′ 8″.5 S. The calculated distance is 66° 49′ to the nearest minute, while the Nautical Almanac gives 66° 48′ 30″. The text states that if latitudes and longitudes were given instead of right ascensions and declinations, the formulas would be the same, with a — a’ being the difference of longitudes, and d and d’ being the colatitudes.
Finding Longitude at Sea by Lunar Observations
Connected to the distance formula is the method of finding longitude at sea by lunar observations. Refraction and parallax cause observers at the surface to refer bodies to different places than they would from the centre. The apparent place of the sun’s limb or a star is always higher than its true place, while the moon’s apparent place is always lower, as its refraction is less than its parallax. The Nautical Almanac contains true distances between the moon and sun and certain stars and planets near the ecliptic, calculated for every 3 hours at Greenwich mean time. If the apparent distance is observed elsewhere and cleared of parallax and refraction, and the mean time at that place is known, the difference of meridians and thus the longitude becomes known. Intervals less than 3 hours can be determined by proportion or interpolation.
To clear the apparent distance, the true and apparent altitudes of the centres are used. A formula is given: sin φ = – h / cos H cos H’ … cos b cos (S — A) cos h cos h’ … cos j (H + H’). By substitution and reduction, sin ½D = cos ½(H + H’) cos φ. Thus, with S and φ known, D is found. A rule is then provided for finding the true distance, requiring only Table III of the “Practical Mathematician’s Pocket Guide.” Corrections must be applied for index error, semidiameter, and depression of the horizon for apparent altitudes, and for refraction and parallax for true altitudes. The rule involves adding the apparent altitudes and apparent distance, taking half the sum, finding the difference between this half sum and the apparent distance, adding the logarithmic cosines of the half sum, difference, and true altitudes, subtracting the sum of logarithmic cosines of apparent altitudes, dividing by 2, increasing by 10, subtracting the logarithmic cosine of half the sum of true altitudes, finding the arc of which the remainder is the logarithmic sine, adding the logarithmic cosine of this arc to the logarithmic cosine of half the sum of true altitudes, and the sum minus 10 is the logarithmic sine of half the true distance.
The mean time at Greenwich is found by taking from the Nautical Almanac the two distances between which the true distance falls, finding their difference, and the difference between the true distance and the first distance. Then, as the first difference is to the second, so is 3 hours to the proportional time, which is added to the time of the first distance. The difference between mean time at Greenwich and mean time at the ship (found by a previous article) is the longitude in time; East if ship time is greater, West if less.
An example is given: on June 17th, 1835, in latitude 29° 8′ N., longitude by account 147° 15′ W., about 8 hours, 32 min. A.M., the distance between the nearest limbs of the sun and moon was observed as 92° 30′ 45″, the moon’s upper limb altitude was 39° 1′ 20″, the sun’s lower limb 42° 36′ 20″, and the height of the eye was 18 feet. The estimated mean time at Greenwich was 6 hours, 21 min. P.M., June 17th. The moon’s semidiameter and horizontal parallax from the Nautical Almanac for 1835 were 15′ 17″ and 55′ 28″. The apparent altitudes were 42° 48′ 2″ and 38° 41′ 59″, and the true altitudes were 42° 47′ 7″ and 39° 24′ 5″. The apparent distance of centres was 93° 21′ 48″. The calculation yields a true distance of 92° 43′ 58″. This falls between 92° 53′ 44″ (at 6 hours) and 91° 28′ 10″ (at 9 hours), giving a proportional time of 0 hours, 20 min., 33 sec. Thus, mean time at Greenwich is 6 hours, 20 min., 33 sec., June 17th, or 30 hours, 20 min., 33 sec., June 16th. Mean time at the place of observation is 20 hours, 29 min., 41 sec. The difference is the longitude in time, which converted to degrees makes 147° 43′ West. The rule has the advantages of requiring reference to only one table, and all logarithms taken out are for cosines, and all arcs are less than 90 degrees, making the calculation straightforward and reliable.
Main Arguments
The central argument of the work is that logarithms constitute an indispensable tool for mathematical practice, transforming tedious arithmetic operations into simple additions and subtractions. The introduction establishes this claim through a systematic exposition of the nature of powers and roots, showing how indices can represent the repeated multiplication of a base number. Multiplication and division of numbers correspond to addition and subtraction of their logarithms; involution and evolution correspond to simple multiplications and divisions of indices.
The work argues that the common system of logarithms, based on the number 10, possesses unique advantages because its base is the same as the root of the decimal scale. Increasing or diminishing the index by unity immediately gives the logarithm of a decimal multiple or submultiple. This means that tabulating decimal parts for integers from 1 to 10,000 or 100,000 suffices for innumerable numbers, with indices supplied by simple rules. Any other base would lack these advantages, requiring separate calculation and tabulation of logarithms with indices for fractions, integers, and powers of the base.
The Neperian system, named for John Napier, is identified as theoretically the most natural system, with a base expressible only by an infinite series. The work argues that this system’s peculiarity lies in its modulus being unity, rendering logarithms independent of the base. Napier could construct his logarithms without reference to any assumed base, a remarkable discovery made half a century before the Logarithmic Series was known. The relationship between different systems is established through the proportionality of their moduli.
The work demonstrates that the rules of plane and spherical trigonometry can be derived from fundamental principles. The proposition that sides of a plane triangle are proportional to the sines of opposite angles is demonstrated by circumscribing a circle around the triangle, drawing radii to angles, and perpendiculars to sides. From this foundation, the work derives formulae for the sine and cosine of sums and differences of angles, leading to the analogies that found practical rules for solving triangles.
The astronomical sections argue that careful observation combined with mathematical analysis can determine celestial distances and positions. The horizontal parallax of a heavenly body is related to its distance from Earth through a simple trigonometric relation. The work shows how simultaneous observations at different locations can determine the moon’s horizontal parallax, as demonstrated by the example of Lacaille and Wargentin’s observations in 1751. The distance to the sun is similarly derived from its parallax.
The work argues that longitude at sea can be determined by lunar observations, using the moon’s motion relative to the fixed stars as a celestial clock. The Nautical Almanac contains true distances between the moon and sun and certain stars and planets near the ecliptic, calculated for every three hours at Greenwich mean time. If the apparent distance is observed elsewhere and cleared of parallax and refraction, the difference of meridians and longitude becomes known.
Key Concepts
The concept of logarithms is developed systematically from first principles. A logarithm is defined as the index of that power of a base equal to a given number. The work traces the etymology of the term to two Greek words meaning indices of numbers, and defines logarithms as a system of artificial numbers indicating or representing natural numbers. The concept is grounded in the nature of powers and roots, with indices indicating how often a root enters as a factor.
The common system of logarithms uses 10 as its base because it is the root of the decimal scale. Powers of 10 are unity with ciphers annexed: 10⁰ equals 1, 10¹ equals 10, 10² equals 100, and so on. Dividing unity by successive natural numbers gives decimal fractions whose logarithms are negative. The logarithm of every proper fraction is essentially negative, increasing numerically as the fraction decreases; the logarithm of 0 is negative infinity.
The Neperian system uses a base of 2.718281828459, expressible only by the infinite series 1 + 1/1 + 1/(1·2) + 1/(1·2·3) + 1/(1·2·3·4) + and so on. The system is sometimes improperly called hyperbolic. Its peculiarity is that its modulus is unity, rendering logarithms independent of the base. The modulus is a constant function of the base, and logarithms of the same number in different systems are proportional to their moduli. The common modulus is the reciprocal of the Neperian logarithm of its base.
The Logarithmic Series is the foundation of logarithm construction. A number’s logarithm in every system is the same infinite series united to a factor called the modulus. The work provides a verbal rule for constructing either Neperian or common logarithms, universally applicable and converging more rapidly as the given number increases. The rule involves dividing unity by the difference between double the given number and unity, dividing successive quotients by the square of that difference, dividing these quotients by odd numbers, summing, doubling, and adding the Neperian logarithm of the difference between the given number and unity.
The sine series is presented as sin x = x(1 — x²/2·3 + x⁴/2·3·4·5 — and so on). Setting this series equal to zero, the values 0, π, 2π, 3π, and their negatives are the roots. The series can be factored as x(1 — x²/π²)(1 — x²/4π²)(1 — x²/9π²) and so on. Taking logarithms of the factors yields a logarithmic series expansion useful for calculating logarithmic sines.
Napier’s Rules for Circular Parts are presented for solving right-angled spherical triangles. The circular parts are the legs, the complement of the hypotenuse, and the complements of the oblique angles. The sine of the mean part equals the sum of the tangents of adjacent parts, or the cosines of remote parts, minus 10. Napier’s Analogies provide six rules relating tangents of half-sums and half-differences of angles and sides, and sines of sides and angles, for solving oblique spherical triangles.
Horizontal parallax is defined as the angle subtended by Earth’s radius at a heavenly body on the horizon. The relation R : sin H :: D : T gives D sin H = T, where H is the horizontal parallax, D the distance, and T Earth’s radius. Parallax in altitude is the difference between zenith distances measured from the surface and from the centre, always less than horizontal parallax and vanishing at the zenith. It equals horizontal parallax times the sine of zenith distance or cosine of altitude.
The difference between true and apparent level is derived from the relation 2TA + A² = d², where T is Earth’s radius, A the height, and d the distance. Omitting A² as negligible gives A = ⅔d² feet for d in miles. The dip of the horizon is the angle between the true horizon and the observed horizon from an elevated position, related to the distance of the visible horizon by tan Dip = d/T.
Themes
The work embodies the Enlightenment ideal of systematic knowledge applied to practical problems. The progression from theoretical foundations to practical applications reflects a conviction that understanding the principles underlying mathematical operations makes their application more reliable and more flexible. The investigations of rules are not mere academic exercises but are intended to equip practitioners with the ability to adapt methods to novel situations.
The theme of intellectual heritage is prominent. The work acknowledges Napier’s discovery of logarithms, noting that he could construct his system without reference to any assumed base, a remarkable achievement made half a century before the Logarithmic Series was known. The author’s careful explanation of his “freedom” in deriving the word “logarithms” to convey its modern acceptation shows an awareness of historical accuracy and a desire to respect the contributions of predecessors while adapting their work to contemporary needs.
The relationship between observation and calculation is a recurring concern. The astronomical sections emphasize that accurate observations must be corrected for various sources of error before they can be used in calculations. Refraction and parallax cause observers at the surface to refer bodies to different places than at the centre. The apparent place of the sun’s limb or a star is always higher than its true place; the moon’s apparent place is always lower, its refraction being less than its parallax. The work acknowledges that refraction depends on air temperature and quotes Professor Woodhouse’s remark that exact formulae are “tools finer than the work.”
The practical orientation of the work reflects the needs of commerce and navigation in the period. Examples involving the cost of goods, the calculation of salaries, the measurement of fields, and the determination of longitude at sea all point to the intended audience of merchants, surveyors, and mariners. The work is not abstract mathematics but mathematics as an instrument of trade and exploration.
The theme of approximation and practical accuracy runs throughout. The work acknowledges that some formulae are only nearly correct if extended further, but sufficiently accurate for practice. The discussion of error analysis using Taylor’s Theorem shows that errors in measurement can be minimized by choosing favorable observing conditions. The example of the dip of the horizon notes that tabular dips differ from truth, especially in the tropics, so navigators use the Dip Sector.
The international character of scientific inquiry is evident in the examples chosen. Observations from the Cape of Good Hope, Stockholm, and ships at sea illustrate the global reach of astronomical and navigational practice. The Nautical Almanac provides data calculated for Greenwich mean time, establishing a universal reference point for navigators of all nations.
The Author’s Conclusions
The work concludes that logarithms, properly understood and applied, provide a powerful and versatile tool for mathematical practice across a wide range of disciplines. The systematic exposition of the nature of logarithms, from the simplest powers and roots to the sophisticated Neperian system, establishes a foundation upon which all subsequent applications rest. The author demonstrates that the common system’s advantage lies in its base being the same as the root of the decimal scale, making it the most convenient for practical calculation.
The investigations of trigonometric formulae establish that the rules for solving plane and spherical triangles are not arbitrary but derive from fundamental principles. The proposition that sides of a plane triangle are proportional to the sines of opposite angles is demonstrated geometrically, and from this foundation the entire edifice of trigonometric solution is constructed. The author shows that Napier’s Rules for Circular Parts and his Analogies provide elegant and efficient methods for solving spherical triangles.
The astronomical sections demonstrate that careful observation combined with mathematical analysis can determine celestial distances and positions with remarkable accuracy. The horizontal parallax of the moon is determined from simultaneous observations at different locations, yielding a value of 60′ 35½” for the specific observation cited, varying between extremes of 61′ 24″ and 53′ 48″. The mean distance to the moon is calculated as 238,603 miles, and the mean distance to the sun as 95,129,500 miles.
The method for finding longitude at sea by lunar observations is presented as a complete and practical procedure. The rule requires reference to one table only, and all logarithms taken out are for cosines. The worked example demonstrates the entire process from raw observations to final longitude, yielding a result of 147° 43′ West for the specific case presented. The author emphasizes the advantages of the method: simplicity, economy of reference, and consistency of logarithmic operations.
The work concludes with the practical tables that enable the methods to be applied without extensive calculation. These tables, retained from the Practical Mathematician’s Pocket Guide, provide the data necessary for surveyors, mariners, astronomers, and engineers to apply the rules and formulae developed in the body of the work. The final example, showing the calculation of the distance between the sun and moon from data in the Nautical Almanac for 1838, demonstrates the integration of tabular data with computational methods that characterizes the work as a whole.
Significance
The work represents a comprehensive attempt to make advanced mathematical methods accessible to a broad audience of practitioners. By combining theoretical investigation with practical application, it bridges the gap between academic mathematics and the needs of merchants, surveyors, mariners, and engineers. The systematic organization, from the nature of logarithms through their application to trigonometry, mensuration, and astronomy, provides a complete course of study for the self-directed learner.
The treatment of the Neperian system is historically significant for its clear explanation of the relationship between different logarithmic systems. The identification of the modulus as the key to converting between systems, and the demonstration that the Neperian modulus is unity, provides a conceptual clarity that was not always present in contemporary treatments. The author’s observation that Napier could construct his logarithms without reference to any assumed base, half a century before the Logarithmic Series was known, highlights the remarkable nature of Napier’s achievement.
The astronomical sections preserve valuable examples of observational practice in the eighteenth and nineteenth centuries. The determination of the moon’s horizontal parallax from the observations of Lacaille and Wargentin in 1751, the calculation of the sun’s distance from its parallax, and the worked example of finding longitude by lunar observations all provide insight into the methods and accuracy of astronomical practice in the period. The discussion of errors in the dip of the horizon, including the observation that tabular values differ from truth especially in the tropics, shows an awareness of the limitations of theoretical formulae in practical conditions.
The work’s significance also lies in its demonstration of the power of logarithmic methods. The examples showing multiplication reduced to addition, division to subtraction, involution to multiplication, and evolution to division illustrate the computational revolution that logarithms made possible. The application of these methods to compound interest, annuities, mensuration, and navigation shows the breadth of their utility.
The publication history of the work, appearing in a Glasgow edition of 1839 as part of a series of pocket guides, testifies to the continuing demand for practical mathematical handbooks in the nineteenth century. The advertisements for other publications in the same series reveal a vibrant market for educational materials aimed at working professionals. The success of these guides, with some reaching six editions and 6000 copies, indicates the importance of such works in the dissemination of practical knowledge.
The work’s final example, calculating the distance between the sun and moon from data in the Nautical Almanac for 1838, demonstrates the integration of tabular data with computational methods that characterizes the work as a whole, showing how the theoretical principles, practical rules, and reference tables combine to solve problems of real navigational significance.
