History of Math Test 3

Calculate (for example) log (1.047) accurately to five decimal places using Euler’s technique utilizing power series

ln(1+x)=x-x²/2 +x³/3-….

for log(1.047), set x=.047

Term 1 (x) = .047

Term 2 (-x²/2)= -(.047)²/2=-.00011045

Term 3 (x³/3): (.047)³/3 = .0000346

Term 4: -.0000012

Term 5: .00000005

Add the values together to get .0459289

round to five decimal places to get ln(1.047)=.04593

Introduction to Analysis of the Infinite

Euler’s text, pioneered these "precalculus" topics designed to support advanced analysis

f(x) and e

Euler is credited with formalizing the notation $e$ for the base of the natural logarithm and f(x) for functions, which standardizes how we write these series today.

Calculate sin(pi/5) accurately to five decimal places using Euler’s technique utilizing power series.

sin(x)=x-x³/3!+x^5/5!-x^7/7!+…

x=pi/5

Term 1: pi/5 = .628315

Term 2: -(pi/5)³/6 = -.0413417

Term 3: (pi/5)^5/5! = .0008165

Term 4: -(pi/5)^7/7! = -.0000077

Term 5: (pi/5)^9/9! = .00000004

Add the values together to get .58779

Calculate cos(pi/5) accurately to five decimal places using Euler’s technique utilizing power series.

cos(x)=1-x²/2!+x^4/4!-x^6/6!

Let x=pi/5

Term 1: 1

Term 2: -(pi/5)²/2 = -.19739209

Term 3: (pi/5)^4/4!= .00649394

Term 4: -(pi/5)^6/6! = -.00008546

Term 5: (pi/5)^8/8! = .0000006

Add together to get .80902

Demonstrate Euler’s method for devising the Power Rule

Define the differential as the difference between the function at x+dx and the function at x. So you y=x^n

dy=(x+dx)^n-x^n

Expand using the bionomial theorem

dy = (x^n+nx^(n-1)dx+(n(n-1)/2)x^(n-2)dx²+…)-x^n

dy=nx^(n-1)dx +(n(n-1)/2)x^(n-2)dx²+…

Since dx is an infinitely tiny value, higher-order powers of it would vanish. So the expression reduces to

d(x^n) = nx^(n-1)dx

Complete a calculation in the style of Jakob Bernoulli for the Law of Large Numbers that determines the number of trials of an experiment (for example, with 100 possible outcomes: 40 “successful” ones, 60 “unsuccessful” ones) that will be necessary to have “moral certainty” that our experimental probability is within 0.1 percent of the theoretical probability Complete a calculation in the style of Jakob Bernoulli for the Law of Large Numbers that determines the number of trials of an experiment (for example, with 100 possible outcomes: 40 “successful” ones, 60 “unsuccessful” ones) that will be necessary to have “moral certainty” that our experimental probability is within 0.1 percent of the theoretical probability

Outcomes

r=40 (successes)

s=60 (failures)

Total, t=100

Theoretical probability r/t=.4

Moral Certainty - Probability of 99.9% or greater, c greater than or equal to 1000 ( the outcome is 1000 times more likely to be within the range than outside of it.

Precision - within .1% of the theoretical probability

Number of trials N must be larger than certainty c, and precision E.

For 50 possible outcomes (r=30, s=20) and precision of 1/50 (.02) The minimum N is 25,550 trials

With 500 outcomes, r=300, s=200 so N goes to 3,149,821 trials

Moral Certainty

It is a practical form of complete conviction regarding an event, defined as having a probability greater than 99.9% (or less than 0.1% for "impossible" situations)

How does the number of repetitions affect moral certainty

The measure of certainty increases toward 99.9% as the number of repetitions increases

How did de Moivre improve upon the law of Large numbers

De Moivre developed a method using the normal curve and integration, showing that far fewer experiments are needed to achieve the same desired accuracy

f the probability of a successful outcome in an experiment is 1/8, find the probability of getting (for example) between 2 and 4 (inclusive) successes in 10 trials of the experiment

Bernoulli trial with success probability p and failure probability q (q=1-p) the probability of achieving exactly k successes in n repeated trials is

P(k) = p^kq^(n-k) (n choose K)

n=1-

p=1/8

q=7/8

k={2,3,4]

Probability of 2 Successes

10 choose 2 = 10!/(2!(10-2)!) = 45

p^k=(1/8)²

q^(n-k)=(7/8)^8

P(k)=.2417

Probability of 3 successes

P(k)=.0921

Probability of 4 Successes

P(k)=.0230

Add together to get .3568

there is approximately 35.68% chance of getting between 2 and 4 successes in 10 trials.

Determine the appropriate division of the stakes in (for example) a best-of-seven series of games
between two unevenly matched players if it were interrupted after the dominant player (who has a
2/3 probability of winning a typical game) has won two games and the other player (who has a
1/3 probability of winning a typical game) has won one

Player A - 2 wins, needs 2 more

Player B - 1 win, needs 3 more

P(A)-2/3

P(B)=1/3

For Player B to win they must win 3 games before A wins 2

Scenario 1 - Win out

(1/3)x(1/3)x(1/3) = 1/27

Scenario 2 - Player B wins 3 out of the next 4 rounds

(2/3)x(1/3)x(1/3)x(1/3) = 2/81

Since there are 3 ways that this could occur - 3(2/81) = 2/27

Probability for Player B is 1/27 + 2/27 = 1/9

Since the game must end with one of them winning, A’s probability of winning is the complement of B’s so

1-1/9 = 8/9

So Player A should receive 8/9 of the winnings and Player B should receive 1/9 of the winning

Determine how many throws of a pair of dice are necessary to ensure at least even odds of rolling doubles (any number) at least once

36 possible outcomes

6 desired outcomes

Probability of success 6/36 = 1/6

Probability of failure 5/6

Smallest n such that

1-(q)^n is greater than or equal to .5

so 1-(5/6)^n

Solve for n to get nln(5/6) is less than or equal to ln(.5)

So n is greater than or equal to 3.801

since n must be a whole number of throws, n=4

So with 4 rolls, the probability of rolling doubles at least once is 1-(5/6)^4 = 51;77%

In a lottery for which the ratio of the number of losing tickets to the number of winning tickets is 49:1, how many tickets should one buy in order to give oneself at least even odds of winning a prize

Probability of winning 1/50 = .02

Probability of losing 49/50 = .98

1-(.98)^n is greater than or equal to .5

solve for n to get n is greater than or equal to 34.31 so you need 35 trials to get a 50.6 % change of winning at least 1 prize

For a lottery in which five balls from a set numbered (for example) between 1 and 60 are drawn, find the odds against a player winning a bet to choose one, two, and three winning numbers

Ways to choose 5 balls from 60 - 60 choose 5

60!/5!55! = 5,461,512

Choosing exactly 1 winner 5 choose 1 - 5

Ways to pick losing balls 55 choose 4 - 341,055

Multiple successful combinations 5×341055 = 1705275

Odds against: (5461512-1705275)/1705275 =2.2 to 1 odds

Choosing exactly 2 winners 5 choose 2 - 10

Ways to pick losing balls 55 choose 3 - 26235

10×26235 = 262350

Odds against (5461512-262350)/262350 = 19.8 to 1

Choosing exactly 3 winners 5 choose 3 - 10

Ways to pick losing balls 55 choose 2 - 1485

10×1485= 14850

Odds against: (5461512-14850)/14850 = 366.8 to 1

Use Bayes’ Theorem to find the probability of a certain event occurring given that one event has already occurred.

$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$

$P(A|B)$ - the probability that event A occurs given B is true

P(B|A)The probability of B occurring given that A is true

P(A) initial estimate of the probability of A before seeing the evidence

P(B) - the evidence/ total probability that B occurring under all possible scenarios

Example:

Imagine a rare disease affects 1% of the population A medical test for this disease has a 95% accuracy rate for those who have it and a 5% false positive rate for those who don't.

If you take the test and it comes back positive, what is the probability that you actually have the disease?

P(A) = .01 = P(disease)

P(B|A) = 0.95 = P(positive given disease)

P(No disease) = .99

P(Positive given no disease) .05

To find P(B) - (.950).01+ .05(.99) = .05

P(disease given positive) - (.95).01/.059 = .161

16.1% chance that you have the disease if you have a positive test result.

Determine the quadratic residues modulo 13 (for example)

Looking for all integers x such that x²=a (mod 13) has a solution. - Squaring integers from 1 to 12, find the remainders when divided by 13

x=1 - x²=1 so x² mod 13 = 1

x=2 - x² mod 13 =4

x=3 x² mod 13 =9

x=4 - x² mod 13 = 3

x=5 - x² mod 13 - 12

x=6 - x² mod 13 = 10

x=7 - x² mod 13 = 10

x=8 - x² mod 13 = 12

x=9 - x² mod 13 = 3

x=10 - x² mod 13 = 9

x=12 - x² mod 13 = 1

Set of quadratic residues - {1,3,,4,9,10,12}

For any prime p, there are exactly (p-1)/2 non-zero quadratic residues. For 13, there is 6.

Show that the expression ab + cd + ef takes on only 15 distinct values despite the 720 potential permutations of the six (different) elements involved

Six elements {a,b,c,d,e,f}

Two ways that the value of ab+cd+ef is the same when a swap is made

Swapping within a pair - (ab=ba)

• Since there are 3 pairs - 2×2×2 = 8 ways to rearrange the elements this way

Swapping the Pairs themselves: - (ab+cd=cd+ab)

• There are 3! ways to arrange the three pairs to get 6

To find the total number of permutations that result in the exact same expression - 8×6=48

Divide the total possible permutations by the ones that take on the same value - 6!/48 = 820/48 = 15

Factor x^7-1 into linear and real quadratic factors using Euler’s method

x^7-1=0 has 7 roots

xk = cos(2pik/7)+isin(2pik/7), for k=0,1,2,3,4,5,6

The root at k=0 is cos(0)+isin(0)=1

The remaining roots are complex conjugates. Euler showed that multiplying the factors of two conjugate roots (x-xk) and (x-x̄k) results in a real quadratic factor of the form

x²-2cos(2pi/7)+1

By grouping pairs (k=1, 6; k=2,5; k=3,4)

For k=1: x²-2cos(2pi/7)x+1

For k=2: x²-2cos(4pi/7)x+1

For k=3: x²-2cos(6pi/7)x+1

So x^7-1= (x-1)(x²-2cos(2pi/7)x+1)(x²-2cos(4pi/7)x+1)(x²-2cos(6pi/7)x+1)

Show the derivation of Maclaurin’s version of Cramer’s Rule

1. ax+by+cz=m

2. dx+ey+fz=m

3. gx+hy+iz=p

Eliminate z to create a two variable system

Multiply equation 1 by f and equation 2 by c to get

afx+bfy+cfz=mf and cdx+cey+cfz=cn

Subtract them to get mf-cn

Multiply equation 1 by i and equation 3 by c

aix+biy+ciz=mi

cgx+chy+ciz=cp

subtract them to get mi-cp

(bi-ch)(mf-cn)

(bf-ce)(mi-cp)

Then the denominator is aie-afh+bfg-bdi+cdh-ceg

With interest at 5%, what is the present value for a life annuity of $30,000 per year for someone of age 36, if their expected maximum age is 86? (What would be a “fair price” to charge for it?

A=p(1-(1+r)^n)/r

P=30000

r=.05

n=86-36=50

Use Euler’s method to decompose a given rational integrand and evaluate it

Example:

Integrate 1/(x²-1)dx

Factor the denominator Q(x) = (x-1)(x+1)

Decompose into partial fractions: =A/(x-1) + B/(x+1)

A=1/2, B=-1/2

So integrate 1/2(x-1) and -1/2(x+1) to get ½ ln (x-1)- ½ ln(x+1) +C

Demonstrate Euler's method for devising the quotient rule for derivatives

Euler considered a quantity y defines as the ration of two functions u and v

y=u/v

yv=u

d(yv)=du using the product rule to get

vdy+ydv=du

solve for dy

dy= (du-ydv)/v substitute back in for y

dy= (du-u/vdv)/v simplifying to get

dy=(vdu-udv)/v²

Jakob Bernoulli’s Important Contributions

• The Law of Large Numbers: He formulated the first version of this fundamental theorem in his work Ars Conjectandi, proving that as the number of trials increases, the observed frequency of an event converges to its actual probability.

• Summation of Power Series: He is known for the "Bernoulli Numbers," which he derived to create a general formula for the sum of the k-th powers of the first n integers.

• Isochrone Curve: He solved the problem of the "isochrone," determining the specific curve along which a particle will fall at a constant vertical velocity.

Johann Bernoulli’s contributions

• L'Hôpital's Rule: Although the rule bears another's name, Johann was the mathematician who actually discovered the method for finding limits of indeterminate forms and sold the discovery to the Marquis de l'Hôpital.

• The Brachistochrone Problem: He famously challenged the mathematical community to find the "curve of quickest descent" between two points, a problem that helped give rise to the calculus of variations.

• Bernoulli Numbers

• Isochrone, catenary

Brook Taylor’s Contributions

• Taylor Series: He is most famous for the "Taylor's Theorem," which allows for the expansion of a function into an infinite sum of terms calculated from the values of its derivatives at a single point.

• Method of Finite Differences: Taylor was a pioneer in this field, which laid the groundwork for the numerical analysis used to approximate solutions to differential equations.

Thomas Simpson’s Contributions

• Numerical Integration (Simpson's Rule): He is recognized for "Simpson's Rule," a method for approximating the definite integral of a function by using parabolic arcs.

• Probability and Error Theory: Simpson was one of the first to apply probability to the theory of errors in physical observations, arguing that the mean of several observations is more reliable than a single one.

Bishop George Berkley’ Contributions

• The Analyst (1734): Berkeley is famous for this critical work, in which he attacked the logical foundations of the calculus developed by Newton and Leibniz.

• "Ghosts of Departed Quantities": He famously mocked the concept of infinitesimals (fluxions), arguing that they were neither finite quantities nor nothing at all, which forced later mathematicians to seek more rigorous foundations for calculus.

Ars Conjectandi

1713

• Foundation of Probability: This posthumously published work by Jakob Bernoulli is considered the founding text of modern probability theory.

• Moral Certainty: It introduced the concept of "moral certainty," distinguishing between absolute mathematical proof and the high degree of probability needed for practical decision-making in life.

• Permutations and Combinations: It provided the first systematic treatment of the theory of combinations and its application to games of chance.

Colin Maclaurin’s contributions

• Treatise of Algebra (1748): This posthumously published work contained the first systematic derivation of what is now known as "Cramer's Rule" for solving systems of linear equations.

• Geometric Treatise on Fluxions: He wrote a major defense of Newton’s calculus (fluxions) in response to Bishop Berkeley’s criticisms, aiming to provide a more rigorous geometric foundation for the method.

• Maclaurin Series: He is known for the special case of the Taylor series centered at zero, which he used to expand functions into infinite power series.

L’Hopital’s Contributions

• Analyse des Infiniment Petits (1696): He authored the first textbook ever written on differential calculus, which helped spread the Leibnizian version of calculus across Europe.

• L'Hôpital's Rule: He is famously associated with the rule for finding the limits of indeterminate forms though it was actually discovered by Johann Bernoulli, who was under contract to provide his discoveries to the Marquis.

• Systematization of Calculus: His work was significant for organizing the diverse techniques of the era into a coherent educational format for the first time.

Maria Gaetana Agnesi’s contributions

• Analytical Institutions (Instituzioni analitiche): Published in 1748, this was the first comprehensive textbook to integrate differential and integral calculus, and it was highly praised for its clarity and organization.

• The "Witch" of Agnesi: She is well-known for the cubic curve a³/(x²+a²)

• . The name "witch" resulted from a mistranslation of the Italian word versiera (meaning "to turn" or "curve") as avversiera ("wife of the devil" or "witch").

• First Female Professor: She was appointed as a professor of mathematics at the University of Bologna by Pope Benedict XIV, recognizing her immense contributions to the field of mathematical analysis.

Euler’s Contributions

• Institutiones calculi differentialis (1755) and Institutiones calculi integralis (1768): These works transformed calculus from a collection of geometric methods into a systematic branch of analysis.

• Factorization of Cyclotomic Polynomials: He developed the method for factoring expressions like x^n - 1 into real linear and quadratic factors using trigonometry and complex roots.

• Modern Mathematical Notation: Euler introduced or popularized much of our standard notation, including e for the base of the natural logarithm, i for the imaginary unit, and the use of $(x) for functions.

Jean le Rond d’Alembert’s contributions

• The D'Alembert Operator: He made significant contributions to the study of partial differential equations, particularly in the context of the vibrating string problem.

• Encyclopédie: As a co-editor with Diderot, he wrote the "Preliminary Discourse" and many mathematical entries, helping to synthesize and spread Enlightenment scientific knowledge.

• D'Alembert's Ratio Test: He is known for this fundamental test used to determine the convergence or divergence of an infinite series

• First coined the term limit

• Solved a wave equation describing the shape of a vibrating string

Joseph-Louis Lagrange’s contributions

• Mécanique Analytique (1788): This landmark work reduced the science of mechanics to a series of algebraic operations, famously boasting that it contained no geometric diagrams.

• Calculus of Variations: He was a primary developer of this field, which seeks to find the function that minimizes or maximizes a specific integral.

• Lagrange Multipliers: He introduced this powerful method for finding the local maxima and minima of a function subject to equality constraints.

• Work with solving quintic equations

• Notation for f’, f’’, f’’’, … for successive derivatives

Abraham de Moivre

• The Doctrine of Chances (1718): This was one of the most influential early works on probability theory, introducing the concept of the normal distribution as an approximation to the binomial distribution.

• Annuities upon Lives (1725): He pioneered the mathematical study of life contingencies, using mortality data to derive formulas for the "fair price" of life annuities.

• De Moivre's Formula: He is famous for the formula that links trigonometry and complex numbers

Thomas Bayes’ contributions

• Bayes' Theorem: He developed a method for "inverse probability," which allows one to calculate the probability of a cause based on an observed effect.

• Posthumous Publication: His landmark paper, An Essay towards solving a Problem in the Doctrine of Chances (1763), was published after his death by Richard Price.

• Foundation of Statistics: His work laid the groundwork for Bayesian statistics, which treats probability as a measure of belief or certainty rather than just frequency.

Pierre-Simon Marquis de Laplace’s contributions

• Théorie analytique des probabilités: He systematized and significantly expanded the work of his predecessors, transforming probability into a rigorous mathematical discipline.

• Celestial Mechanics: Known as the "Newton of France," he proved the stability of the solar system using mathematical analysis in his multi-volume Traité de mécanique céleste.

• Laplace’s Equation: He developed this fundamental partial differential equation, which is essential for studying electromagnetism, astronomy, and fluid dynamics.

Arithmetica Universalis

1707

• Newton’s Lectures: This work was based on Isaac Newton’s lectures on algebra and was published by William Whiston.

• Algebra as Generalized Arithmetic: It promoted the idea that algebra is a "universal arithmetic" where letters represent generalized numbers, helping to standardize algebraic notation.

• Rule of Signs: It contained Newton's version of the rule for determining the number of imaginary roots in a polynomial equation.

Treatise of Algebra

1748

• Colin Maclaurin: This influential textbook was published posthumously and became a standard for teaching algebra in the 18th century.

• Elimination Methods: It provided the first systematic explanation of solving systems of linear equations using the method of elimination, which we now recognize as Cramer’s Rule.

• Geometric Foundations: Maclaurin used this work to attempt to provide a more rigorous, quasi-geometric foundation for algebraic operations.

Complete introduction to Algebra

1770

• Leonhard Euler: This is one of the most famous and widely translated algebra textbooks in history, written by Euler after he had become blind.

• Pedagogical Clarity: It was celebrated for its clear, step-by-step instructions, making complex algebraic concepts accessible to students.

• Diophantine Equations: The work concludes with a significant section on "Indeterminate Analysis," providing methods for solving various Diophantine equations.

Reflection on the Theory of Euqations

• Joseph-Louis Lagrange: This influential memoir was written by Lagrange and served as a critical turning point in the study of algebraic equations.

• Permutations of Roots: Lagrange investigated why the general methods for solving cubic and quartic equations worked, discovering that the key lay in the permutations of the roots.

• Foundation for Group Theory: While he did not solve the quintic equation, his systematic analysis of "resolvents" and the symmetry of roots laid the essential groundwork for the future work of Abel and Galois in group theory.

Alexis-Claude Clairaut’s Contributions

• Calculus of Variations: He was a child prodigy who made early, significant contributions to the development of the calculus of variations alongside Euler.

• Clairaut's Equation: He is known for the differential equation of the form y = xy' + f(y'), which has both a general linear solution and a singular solution.

• Shape of the Earth: He published Théorie de la figure de la terre in 1743, which used mathematical analysis to confirm Newton's theory that the Earth is an oblate spheroid.

Parallel Postulate

• Euclid's Fifth Postulate: This postulate states that if a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side.

• The "Flaw" in Geometry: Many 18th-century mathematicians considered this postulate less "self-evident" than the others and attempted to prove it as a theorem derived from the first four postulates.

• Stimulus for Discovery: These failed attempts to prove the postulate eventually led to the discovery of non-Euclidean geometries in the 19th century.

Giovanni Girolamo Saccheri’s Contributions

• Euclides ab omni naevo vindicatus (1733): He authored this work, whose title translates to "Euclid Cleared of Every Flaw," in an attempt to prove the parallel postulate through a reductio ad absurdum.

• The Saccheri Quadrilateral: He analyzed a quadrilateral with two equal sides perpendicular to a base, examining the "Hypothesis of the Acute Angle" and the "Hypothesis of the Obtuse Angle".

• Unwitting Pioneer: Although he believed he had successfully defended Euclid by "proving" the acute angle hypothesis led to a contradiction, his detailed theorems actually described the first consistent results of what would later be known as hyperbolic geometry.

Johann Lambert’s contributions

• Irrationality of pi: In 1761, he provided the first rigorous proof that pi and e are irrational numbers, using continued fractions to demonstrate that they cannot be expressed as a ratio of two integers.

• Hyperbolic Trigonometry: He was a pioneer in developing the theory of hyperbolic functions (sinh, cosh, tanh), linking them to the geometry of the hyperbola in the same way circular functions link to the circle.

• Non-Euclidean Geometry: Building on the work of Saccheri, Lambert investigated the "Hypothesis of the Acute Angle" and noted that a geometry based on this hypothesis would occur on a sphere of imaginary radius.

Bridges of Konigsberg

• Leonhard Euler (1736): Euler famously solved this puzzle, which asked if it was possible to walk through the city of Königsberg crossing each of its seven bridges exactly once.

• Birth of Graph Theory: Euler’s solution ignored the physical distances and focused only on the connections between landmasses, creating the first formal application of what we now call Graph Theory.

• Vertex Degrees: He proved that a path is only possible if the "degree" (number of edges) of each vertex is even, or if exactly two vertices have odd degrees.

Gaspard Monge’s contributions

• Descriptive Geometry: He is considered the father of descriptive geometry, a method for representing three-dimensional objects in two dimensions using orthographic projections, which became essential for engineering and military architecture.

• Differential Geometry: Monge made significant contributions to the study of curved surfaces and "lines of curvature," applying calculus to solve complex geometric problems.

• Educational Reform: He was a founding member of the École Polytechnique in France, which revolutionized how mathematics and engineering were taught to future generations of scientists.

Jean-Charles Chevalier de Borda’s contributions

• The Borda Count: He developed a preferential voting system (the Borda count) where voters rank candidates in order of preference, designed to find a consensus candidate rather than just a simple majority winner.

• Metric System Development: Borda was a key member of the commission that established the metric system, specifically contributing to the measurement of the meridian arc to define the meter.

• Experimental Physics: He improved the design of the "repeating circle," a surveying instrument that allowed for much more precise angular measurements in geodesy and navigation.

Jakob Bernoulli

Switzerland

Older brother

chair at the university of basel

Johann Bernoulli

Switzerland

Teacher - students included Euler and l’hopital

younger brother

Brook Taylor

1685-1731

Edmonton, Middlesex, England

Defender of Newton’s Fluxions

Secretary of the royal society

Thomas Simpson

Market Bosworth, Leicestershire, England

Self-taught

Professor at royal military academy of woolwich

Bishop George Berkely

1685-1753

Kilkenny, ireland

Philosopher and anglican bishop of cloyne

Colin Maclaurin

Kilmodan, scotland

Child prodigy

became a professor at 19

Close friend with Newton

Guillaume de l’Hopital

Paris, France

French nobleman

studied under bernoulli

credited with writing the first textbook on differential calculus

Maria Gaetana Agensi

Milan, Italy

Linguistic and mathematical prodigy

First woman to write a math textbook

professor at the university of bologna

Leonhard Euler

Basel, Switzerland

Became completely blind as he got old

Jean le Rond d’alembert

Paris, France

philosopher, physicist, and mathematician of the Enlightenment

co-editor of Encyclopedie

Joseph-Louis Lagrange

Turin, Italy

Central figure of the Enlightenment

Succeeded Euler as the director of mathematics at the Berlin Academy

Senator in Napoleonic France

Abraham de Moivre

Vitry-le-Francois, France

French Hugenout who fled england to escape religious persecution

Close associate of Newton

Member of the royal society

Thomas Bayes

London, England

Presbyterian Minister and mathematician

Most famous work was published Posthumously

Pierre-Simon, Marquis de Laplace

Normandy, France

Called the “Newton of France”

Became a count and then marquis after the French Revolution

Alexis-Claude Clairaut

Paris, France

Child prodigy

Key figure in the French Newtonian movement

Giovanni Girolamo Saccheri

Sanremo, Italy

Jesuit priest and professor of mathematics at the university of Pavia

brilliant logician who sought to defend the geometric foundations laid by Euclid

Johann Heinrich Lambert

Mulhouse (Switzerland during his time, now France)