Part IV: Early Modern Transformation

Viète was a French lawyer whose spare-time mathematics revolutionized symbolic notation. In his In Artem Analyticem Isagoge (1591), he introduced the systematic use of letters for both known quantities (consonants B, C, D…) and unknowns (vowels A, E, I…) — finally separating the general statement of an algebraic problem from its particular numerical instance.

His species logistic (symbolic calculation) allowed mathematicians to manipulate equations in their general form for the first time. He also discovered the first infinite product for π: 2/π = √½ · √(½+½√½) · √(½+½√(½+½√½)) · …

Viète used his analytical methods to break a Spanish diplomatic cipher used by Philip II of Spain, demonstrating that mathematical abstraction had real-world applications far beyond geometry.

A Scottish laird with a passion for astronomy and theology, Napier spent 20 years developing logarithms — published in his Mirifici Logarithmorum Canonis Descriptio (1614). Logarithms transformed multiplication and division into addition and subtraction, reducing the labor of astronomical calculation by orders of magnitude.

The key insight: if a = b^n and c = b^m, then ac = b^(n+m). By tabulating logarithms, one need only look up values and add them. The astronomer Johannes Kepler immediately used Napier's tables to check his own planetary calculations.

Henry Briggs visited Napier and together they agreed to base the system on 10 (common logarithms), which Briggs published in 1617. Napier also invented Napier's bones — physical rods for performing multiplication mechanically.

A Flemish engineer and mathematician, Stevin published De Thiende (The Tenth, 1585), which introduced and championed decimal fractions. Before Stevin, fractions were always expressed as ratios of integers; Stevin showed that writing 3.14 was not only valid but far more convenient for practical calculation.

Stevin explicitly advocated that governments adopt a decimal currency, weights, and measures — an idea not widely implemented until the French Revolution introduced the metric system two centuries later. He also contributed to hydrostatics, showing that the pressure in a fluid depends only on depth, not on the shape of the container.

An Italian Jesuit priest and student of Galileo, Cavalieri developed the method of indivisibles in Geometria Indivisibilibus (1635). His principle: if two solids have equal cross-sectional areas at every height, they have equal volumes. In modern notation, this is the precursor to integration.

Cavalieri proved that ∫₀¹ xⁿ dx = 1/(n+1) for positive integers n using geometric arguments, though he lacked the notation to express it this cleanly. This result was crucial to Newton and Leibniz later.

His method was controversial — critics asked: what exactly is an indivisible? Cavalieri avoided the philosophical question by treating his method purely as a computational tool, essentially the same pragmatic attitude that would justify Leibniz's infinitesimals for centuries.

Wallis was an English mathematician who served as cryptographer to Parliament during the Civil War. His Arithmetica Infinitorum (1656) extended Cavalieri's methods using algebraic interpolation and gave the remarkable Wallis product:

π/2 = (2/1) · (2/3) · (4/3) · (4/5) · (6/5) · (6/7) · …

This was one of the first infinite products in mathematics — and the first analytical expression for π. Wallis also introduced the ∞ symbol for infinity, which we still use, and anticipated Newton's binomial theorem by computing fractional powers through interpolation.

Newton credited Arithmetica Infinitorum as the inspiration for his own fluxions. Wallis also wrote the first systematic history of algebra, tracing the subject from ancient times to his day.

Newton's teacher and predecessor as Lucasian Professor of Mathematics at Cambridge, Barrow proved the Fundamental Theorem of Calculus in geometric form in his Lectiones Geometricae (1670) — showing that finding tangent lines (differentiation) and computing areas (integration) are inverse operations.

Barrow did not pursue the algebraic formalism that would make calculus computational; he retained the Greek geometric style. He recognized Newton's superior abilities and resigned his chair in 1669 so that Newton could succeed him — one of the most consequential acts of academic generosity in history.

A Scottish mathematician who died at only 36, Gregory made contributions far ahead of his time. He discovered the series for arctan independently of Leibniz:

arctan x = x − x³/3 + x⁵/5 − x⁷/7 + …

Setting x = 1 gives the Leibniz–Gregory series: π/4 = 1 − 1/3 + 1/5 − 1/7 + … — the first infinite series for π, though it converges too slowly to be practical. Gregory also gave the first clear distinction between convergent and divergent infinite series — a concept not rigorously settled until Cauchy in the 19th century.

Gregory invented the reflecting telescope independently of Newton and proved that the quadrature of the circle (constructing a square of equal area using compass and ruler) is impossible — anticipating the proof of π's transcendence by Lindemann in 1882.

The greatest Dutch scientist of his age, Huygens published Van Rekeningh in Spelen van Geluck(On Calculation in Games of Chance, 1657) — the first published treatise on probability theory, introducing the concept of mathematical expectation (expected value).

Huygens proved: if you have probability p of winning A and (1−p) of winning B, your expected gain is pA + (1−p)B. This abstraction allowed for the first time a rigorous comparison of different gambles.

Beyond probability, Huygens invented the pendulum clock (1656), dramatically improving time measurement; developed the wave theory of light; discovered Saturn's ring and its moon Titan; and made fundamental contributions to mechanics, including centripetal force.

A French Huguenot refugee in London, de Moivre became friends with Newton and was elected to the Royal Society. His greatest contribution to pure mathematics is de Moivre's theorem:

(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)

This elegant result connects trigonometry, complex numbers, and the binomial theorem in a single formula — a cornerstone of complex analysis. In probability, de Moivre derived the normal approximation to the binomial distribution in 1733, anticipating the Central Limit Theorem: for large n, binomial probabilities cluster around a bell-shaped curve.

Legend has it that de Moivre, noting he was sleeping an extra 15 minutes each day, predicted the exact date of his own death by arithmetic progression — and was correct.

A student and successor of Galileo, Torricelli extended Cavalieri's method of indivisibles in remarkable ways. Most famously, he computed the volume of Gabriel's Horn(the solid of revolution of y = 1/x for x ≥ 1):

Volume = π · ∫₁^∞ (1/x)² dx = π < ∞

Yet its surface area is infinite. This “Torricelli's trumpet” paradox — a finite solid you could fill with paint but not coat on the outside — astonished mathematicians of the age and forced a deeper examination of the meaning of infinity.

Torricelli also invented the barometer (1643), demonstrating atmospheric pressure, and made fundamental contributions to fluid dynamics (Torricelli's theorem on efflux velocity).