Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/60² + 10/60³ = 1.41421296...^[1]

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).

One of the earliest mathematical writings is a Babylonian tablet from the Yale Babylonian Collection(YBC 7289), which gives a sexagesimal numerical approximation of Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{2} , the length of the diagonal in a unit square. Being able to compute the sides of a triangle (and hence, being able to compute square roots) is extremely important, for instance, in carpentry and construction.^[2]

Numerical analysis continues this long tradition of practical mathematical calculations. Much like the Babylonian approximation of Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{2} , modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors.

Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21st century, the life sciences and even the arts have adopted elements of scientific computations. Ordinary differential equations appear in the movement of heavenly bodies (planets, stars and galaxies); optimization occurs in portfolio management; numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.

Before the advent of modern computers numerical methods often depended on hand interpolation in large printed tables. Since the mid 20th century, computers calculate the required functions instead. These same interpolation formulas nevertheless continue to be used as part of the software algorithms for solving differential equations.

General introduction[link]

The overall goal of the field of numerical analysis is the design and analysis of techniques to give approximate but accurate solutions to hard problems, the variety of which is suggested by the following.

• Advanced numerical methods are essential in making numerical weather prediction feasible.
• Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations.
• Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving partial differential equations numerically.
• Hedge funds (private investment funds) use tools from all fields of numerical analysis to calculate the value of stocks and derivatives more precisely than other market participants.
• Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments and fuel needs. This field is also called operations research.
• Insurance companies use numerical programs for actuarial analysis.

The rest of this section outlines several important themes of numerical analysis.

History[link]

The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method.

To facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients. Using these tables, often calculated out to 16 decimal places or more for some functions, one could look up values to plug into the formulas given and achieve very good numerical estimates of some functions. The canonical work in the field is the NIST publication edited by Abramowitz and Stegun, a 1000-plus page book of a very large number of commonly used formulas and functions and their values at many points. The function values are no longer very useful when a computer is available, but the large listing of formulas can still be very handy.

The mechanical calculator was also developed as a tool for hand computation. These calculators evolved into electronic computers in the 1940s, and it was then found that these computers were also useful for administrative purposes. But the invention of the computer also influenced the field of numerical analysis, since now longer and more complicated calculations could be done.

Direct and iterative methods[link]

Direct vs iterative methods Consider the problem of solving 3x3 + 4 = 28 for the unknown quantity x. Direct method 3x3 + 4 = 28. Subtract 4 3x3 = 24. Divide by 3 x3 = 8. Take cube roots x = 2. For the iterative method, apply the bisection method to f(x) = 3x3 − 24. The initial values are a = 0, b = 3, f(a) = −24, f(b) = 57. Iterative method a b mid f(mid) 0 3 1.5 −13.875 1.5 3 2.25 10.17... 1.5 2.25 1.875 −4.22... 1.875 2.25 2.0625 2.32... We conclude from this table that the solution is between 1.875 and 2.0625. The algorithm might return any number in that range with an error less than 0.2. Discretization and numerical integration[link] In a two hour race, we have measured the speed of the car at three instants and recorded them in the following table. Time 0:20 1:00 1:40 km/h 140 150 180 A discretization would be to say that the speed of the car was constant from 0:00 to 0:40, then from 0:40 to 1:20 and finally from 1:20 to 2:00. For instance, the total distance traveled in the first 40 minutes is approximately (2/3h × 140 km/h) = 93.3 km. This would allow us to estimate the total distance traveled as 93.3 km + 100 km + 120 km = 313.3 km, which is an example of numerical integration (see below) using a Riemann sum, because displacement is the integral of velocity. Ill posed problem: Take the function f(x) = 1/(x − 1). Note that f(1.1) = 10 and f(1.001) = 1000: a change in x of less than 0.1 turns into a change in f(x) of nearly 1000. Evaluating f(x) near x = 1 is an ill-conditioned problem. Well-posed problem: By contrast, the function Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)=\sqrt{x} is continuous and so evaluating it is well-posed, at least for x being not close to zero.

Direct methods compute the solution to a problem in a finite number of steps. These methods would give the precise answer if they were performed in infinite precision arithmetic. Examples include Gaussian elimination, the QR factorization method for solving systems of linear equations, and the simplex method of linear programming. In practice, finite precision is used and the result is an approximation of the true solution (assuming stability).

In contrast to direct methods, iterative methods are not expected to terminate in a number of steps. Starting from an initial guess, iterative methods form successive approximations that converge to the exact solution only in the limit. A convergence test is specified in order to decide when a sufficiently accurate solution has (hopefully) been found. Even using infinite precision arithmetic these methods would not reach the solution within a finite number of steps (in general). Examples include Newton's method, the bisection method, and Jacobi iteration. In computational matrix algebra, iterative methods are generally needed for large problems.

Iterative methods are more common than direct methods in numerical analysis. Some methods are direct in principle but are usually used as though they were not, e.g. GMRES and the conjugate gradient method. For these methods the number of steps needed to obtain the exact solution is so large that an approximation is accepted in the same manner as for an iterative method.

Discretization[link]

Furthermore, continuous problems must sometimes be replaced by a discrete problem whose solution is known to approximate that of the continuous problem; this process is called discretization. For example, the solution of a differential equation is a function. This function must be represented by a finite amount of data, for instance by its value at a finite number of points at its domain, even though this domain is a continuum.

Generation and propagation of errors[link]

The study of errors forms an important part of numerical analysis. There are several ways in which error can be introduced in the solution of the problem.

Round-off[link]

Round-off errors arise because it is impossible to represent all real numbers exactly on a machine with finite memory (which is what all practical digital computers are).

Truncation and discretization error[link]

Truncation errors are committed when an iterative method is terminated or a mathematical procedure is approximated, and the approximate solution differs from the exact solution. Similarly, discretization induces a discretization error because the solution of the discrete problem does not coincide with the solution of the continuous problem. For instance, in the iteration in the sidebar to compute the solution of Failed to parse (Missing texvc executable; please see math/README to configure.): 3x^3+4=28 , after 10 or so iterations, we conclude that the root is roughly 1.99 (for example). We therefore have a truncation error of 0.01.

Once an error is generated, it will generally propagate through the calculation. For instance, we have already noted that the operation + on a calculator (or a computer) is inexact. It follows that a calculation of the type a+b+c+d+e is even more inexact.

What does it mean when we say that the truncation error is created when we approximate a mathematical procedure? We know that to integrate a function exactly requires one to find the sum of infinite trapezoids. But numerically one can find the sum of only finite trapezoids, and hence the approximation of the mathematical procedure. Similarly, to differentiate a function, the differential element approaches to zero but numerically we can only choose a finite value of the differential element.

Numerical stability and well-posed problems[link]

Numerical stability is an important notion in numerical analysis. An algorithm is called numerically stable if an error, whatever its cause, does not grow to be much larger during the calculation. This happens if the problem is well-conditioned, meaning that the solution changes by only a small amount if the problem data are changed by a small amount. To the contrary, if a problem is ill-conditioned, then any small error in the data will grow to be a large error.

Both the original problem and the algorithm used to solve that problem can be well-conditioned and/or ill-conditioned, and any combination is possible.

So an algorithm that solves a well-conditioned problem may be either numerically stable or numerically unstable. An art of numerical analysis is to find a stable algorithm for solving a well-posed mathematical problem. For instance, computing the square root of 2 (which is roughly 1.41421) is a well-posed problem. Many algorithms solve this problem by starting with an initial approximation x₁ to Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{2} , for instance x₁=1.4, and then computing improved guesses x₂, x₃, etc.. One such method is the famous Babylonian method, which is given by x_k+1 = x_k/2 + 1/x_k. Another iteration, which we will call Method X, is given by x_{k + 1} = (x_k²−2)² + x_k.^[3] We have calculated a few iterations of each scheme in table form below, with initial guesses x₁ = 1.4 and x₁ = 1.42.

Observe that the Babylonian method converges fast regardless of the initial guess, whereas Method X converges extremely slowly with initial guess 1.4 and diverges for initial guess 1.42. Hence, the Babylonian method is numerically stable, while Method X is numerically unstable.

Numerical stability is affected by the number of the significant digits the machine keeps on, if we use a machine that keeps on the first four floating-point digits, a good example on loss of significance is given by these two equivalent functions

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)=x\left(\sqrt{x+1}-\sqrt{x}\right) \text{ and } g(x)=\frac{x}{\sqrt{x+1}+\sqrt{x}}.

If we compare the results of

Failed to parse (Missing texvc executable; please see math/README to configure.): f(500)=500(\sqrt{501}-\sqrt{500})=500(22.3830-22.3607)=500(0.0223)=11.1500

and

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{alignat}{3}g(500)&=\frac{500}{\sqrt{501}+\sqrt{500}}\\ &=\frac{500}{22.3830+22.3607}\\ &=\frac{500}{44.7437}=11.1748 \end{alignat}

by looking to the two above results, we realize that loss of significance which is also called Subtractive Cancelation has a huge effect on the results, even though both functions are equivalent; to show that they are equivalent simply we need to start by f(x) and end with g(x), and so

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{alignat}{4} f(x)&=x(\sqrt{x+1}-\sqrt{x})\\ & =x(\sqrt{x+1}-\sqrt{x})\frac{(\sqrt{x+1}+\sqrt{x})}{(\sqrt{x+1}+\sqrt{x})}\\ &=x\frac{((\sqrt{x+1})^2-(\sqrt{x})^2)}{(\sqrt{x+1}+\sqrt{x})} &=\frac {x}{(\sqrt{x+1}+\sqrt{x})} \end{alignat}

The true value for the result is 11.174755..., which is exactly g(500) = 11.1748 after rounding the result to 4 decimal digits.

Now imagine that lots of terms like these functions are used in the program; the error will increase as one proceeds in the program, unless one uses the suitable formula of the two functions each time one evaluates either f(x), or g(x); the choice is dependent on the parity of x.

• The example is taken from Mathew; Numerical methods using matlab, 3rd ed.

Areas of study[link]

The field of numerical analysis is divided into different disciplines according to the problem that is to be solved.

Computing values of functions[link]

Interpolation: We have observed the temperature to vary from 20 degrees Celsius at 1:00 to 14 degrees at 3:00. A linear interpolation of this data would conclude that it was 17 degrees at 2:00 and 18.5 degrees at 1:30pm. Extrapolation: If the gross domestic product of a country has been growing an average of 5% per year and was 100 billion dollars last year, we might extrapolate that it will be 105 billion dollars this year. Regression: In linear regression, given n points, we compute a line that passes as close as possible to those n points. Optimization: Say you sell lemonade at a lemonade stand, and notice that at $1, you can sell 197 glasses of lemonade per day, and that for each increase of $0.01, you will sell one glass of lemonade less per day. If you could charge $1.485, you would maximize your profit, but due to the constraint of having to charge a whole cent amount, charging $1.49 per glass will yield the maximum income of $220.52 per day. Differential equation: If you set up 100 fans to blow air from one end of the room to the other and then you drop a feather into the wind, what happens? The feather will follow the air currents, which may be very complex. One approximation is to measure the speed at which the air is blowing near the feather every second, and advance the simulated feather as if it were moving in a straight line at that same speed for one second, before measuring the wind speed again. This is called the Euler method for solving an ordinary differential equation.

One of the simplest problems is the evaluation of a function at a given point. The most straightforward approach, of just plugging in the number in the formula is sometimes not very efficient. For polynomials, a better approach is using the Horner scheme, since it reduces the necessary number of multiplications and additions. Generally, it is important to estimate and control round-off errors arising from the use of floating point arithmetic.

Interpolation, extrapolation, and regression[link]

Interpolation solves the following problem: given the value of some unknown function at a number of points, what value does that function have at some other point between the given points?

Extrapolation is very similar to interpolation, except that now we want to find the value of the unknown function at a point which is outside the given points.

Regression is also similar, but it takes into account that the data is imprecise. Given some points, and a measurement of the value of some function at these points (with an error), we want to determine the unknown function. The least squares-method is one popular way to achieve this.

Solving equations and systems of equations[link]

Another fundamental problem is computing the solution of some given equation. Two cases are commonly distinguished, depending on whether the equation is linear or not. For instance, the equation Failed to parse (Missing texvc executable; please see math/README to configure.): 2x+5=3

is linear while Failed to parse (Missing texvc executable; please see math/README to configure.): 2x^2+5=3
is not.

Much effort has been put in the development of methods for solving systems of linear equations. Standard direct methods, i.e., methods that use some matrix decomposition are Gaussian elimination, LU decomposition, Cholesky decomposition for symmetric (or hermitian) and positive-definite matrix, and QR decomposition for non-square matrices. Iterative methods such as the Jacobi method, Gauss–Seidel method, successive over-relaxation and conjugate gradient method are usually preferred for large systems.

Root-finding algorithms are used to solve nonlinear equations (they are so named since a root of a function is an argument for which the function yields zero). If the function is differentiable and the derivative is known, then Newton's method is a popular choice. Linearization is another technique for solving nonlinear equations.

Solving eigenvalue or singular value problems[link]

Several important problems can be phrased in terms of eigenvalue decompositions or singular value decompositions. For instance, the spectral image compression algorithm^[4] is based on the singular value decomposition. The corresponding tool in statistics is called principal component analysis.

Optimization[link]

Optimization problems ask for the point at which a given function is maximized (or minimized). Often, the point also has to satisfy some constraints.

The field of optimization is further split in several subfields, depending on the form of the objective function and the constraint. For instance, linear programming deals with the case that both the objective function and the constraints are linear. A famous method in linear programming is the simplex method.

The method of Lagrange multipliers can be used to reduce optimization problems with constraints to unconstrained optimization problems.

Evaluating integrals[link]

Numerical integration, in some instances also known as numerical quadrature, asks for the value of a definite integral. Popular methods use one of the Newton–Cotes formulas (like the midpoint rule or Simpson's rule) or Gaussian quadrature. These methods rely on a "divide and conquer" strategy, whereby an integral on a relatively large set is broken down into integrals on smaller sets. In higher dimensions, where these methods become prohibitively expensive in terms of computational effort, one may use Monte Carlo or quasi-Monte Carlo methods (see Monte Carlo integration), or, in modestly large dimensions, the method of sparse grids.

Differential equations[link]

Numerical analysis is also concerned with computing (in an approximate way) the solution of differential equations, both ordinary differential equations and partial differential equations.

Partial differential equations are solved by first discretizing the equation, bringing it into a finite-dimensional subspace. This can be done by a finite element method, a finite difference method, or (particularly in engineering) a finite volume method. The theoretical justification of these methods often involves theorems from functional analysis. This reduces the problem to the solution of an algebraic equation.

Software[link]

Since the late twentieth century, most algorithms are implemented in a variety of programming languages. The Netlib repository contains various collections of software routines for numerical problems, mostly in Fortran and C. Commercial products implementing many different numerical algorithms include the IMSL and NAG libraries; a free alternative is the GNU Scientific Library.

There are several popular numerical computing applications such as MATLAB, S-PLUS, LabVIEW, and IDL as well as free and open source alternatives such as FreeMat, Scilab, GNU Octave (similar to Matlab), IT++ (a C++ library), R (similar to S-PLUS) and certain variants of Python. Performance varies widely: while vector and matrix operations are usually fast, scalar loops may vary in speed by more than an order of magnitude.^[5][6]

Many computer algebra systems such as Mathematica also benefit from the availability of arbitrary precision arithmetic which can provide more accurate results.

Also, any spreadsheet software can be used to solve simple problems relating to numerical analysis.

See also[link]

• Scientific computing
• List of numerical analysis topics
• Gram-Schmidt process
• Numerical differentiation
• Symbolic-numeric computation
• Analysis of algorithms
• Numerical Recipes

Notes[link]

1. ^ Photograph, illustration, and description of the root(2) tablet from the Yale Babylonian Collection
2. ^ The New Zealand Qualification authority specifically mentions this skill in document 13004 version 2, dated 17 October 2003 titled CARPENTRY THEORY: Demonstrate knowledge of setting out a building
3. ^ This is a fixed point iteration for the equation Failed to parse (Missing texvc executable; please see math/README to configure.): x=(x^2-2)^2+x=f(x) , whose solutions include Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{2} . The iterates always move to the right since Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)\geq x . Hence Failed to parse (Missing texvc executable; please see math/README to configure.): x_1=1.4<\sqrt{2} converges and Failed to parse (Missing texvc executable; please see math/README to configure.): x_1=1.42>\sqrt{2} diverges.
4. ^ The Singular Value Decomposition and Its Applications in Image Compression
5. ^ Speed comparison of various number crunching packages
6. ^ Comparison of mathematical programs for data analysis Stefan Steinhaus, ScientificWeb.com

References[link]

• Gilat, Amos (2004). MATLAB: An Introduction with Applications (2nd edition ed.). John Wiley & Sons. ISBN 0-471-69420-7. 
• Hildebrand, F. B. (1974). Introduction to Numerical Analysis (2nd edition ed.). McGraw-Hill. ISBN 0-07-028761-9. 
• Leader, Jeffery J. (2004). Numerical Analysis and Scientific Computation. Addison Wesley. ISBN 0-201-73499-0. 
• Trefethen, Lloyd N. (2006). "Numerical analysis", 20 pages. In: Timothy Gowers and June Barrow-Green (editors), Princeton Companion of Mathematics, Princeton University Press.
• Higham, Nicholas J. (1966). Accuracy and Stability of Numerical Algorithms (Society for Industrial and Applied Mathematics, ISBN 0-89871-355-2). 
• Gene F. Golub and Charles F. van Loan (1986). Matrix Computations, Third Edition (Johns Hopkins University Press, ISBN 0-8018-5413). 
• J.H. Wilkinson M.A., D. Sc. (1965)). The Algebraic Eigenvalue Problem (Clarendon Press). 
• Kahan, W. [1972]. “A survey of error-analysis,” in Info. Processing 71 (Proc. IFIP Congress 71 in Ljubljana), vol. 2, pp. 1214–39, North-Holland Publishing, Amsterdam.  (examples of the importance of accurate arithmetic).

External links[link]

Wikibooks has more on the topic of Numerical analysis

Journals

• Numerische Mathematik, volumes 1-66, Springer, 1959-1994 (searchable; pages are images). (English) (German)
• Numerische Mathematik at SpringerLink, volumes 1-112, Springer, 1959–2009
• SIAM Journal on Numerical Analysis, volumes 1-47, SIAM, 1964–2009

Software and Code

• Numerical methods for Fortran programmers
• Java Number Cruncher features free, downloadable code samples that graphically illustrate common numerical algorithms
• Excel Implementations
• Several Numerical Mathematical Utilities (in Javascript)

Online Texts

• Numerical Recipes, William H. Press (free, downloadable previous editions)
• First Steps in Numerical Analysis, R.J.Hosking, S.Joe, D.C.Joyce, and J.C.Turner
• Numerical Analysis for Engineering, D. W. Harder
• CSEP (Computational Science Education Project), U.S. Department of Energy

Online Course Material

• Numerical Methods, Stuart Dalziel University of Cambridge
• Lectures on Numerical Analysis, Dennis Deturck and Herbert S. Wilf University of Pennsylvania
• Numerical methods, John D. Fenton University of Karlsruhe
• Numerical Methods for Science, Technology, Engineering and Mathematics, Autar Kaw University of South Florida
• Numerical Analysis Project, John H. Mathews California State University, Fullerton
• Numerical Methods - Online Course, Aaron Naiman Jerusalem College of Technology
• Numerical Methods for Physicists, Anthony O’Hare Oxford University
• Lectures in Numerical Analysis, R. Radok Mahidol University
• Introduction to Numerical Analysis for Engineering, Henrik Schmidt Massachusetts Institute of Technology

Milton Bradley
Born	(1836-11-08)November 8, 1836 Vienna, Maine, U.S.
Died	May 30, 1911(1911-05-30) (aged 74) Springfield, Massachusetts, U.S.
Occupation	Board game manufacturer

Milton Bradley (November 8, 1836 – May 30, 1911), an American game pioneer, was credited by many with launching the board game industry in North America with Milton Bradley Company.

A native of Vienna, Maine, Bradley grew up in a working-class household in Lowell, Massachusetts. After completing high school he found work as a draftsman before enrolling at the Lawrence Scientific School in Cambridge, Massachusetts. In 1856, he secured employment at the Watson Company in Springfield, Massachusetts. After the company was shuttered during the recession of 1858, he entered business for himself as a mechanical draftsman and patent agent. Later, Bradley pursued lithography and in 1860, he set up the first color lithography shop in Springfield, Massachusetts. Eventually, Bradley moved forward with an idea he had for a board game which he called The Checkered Game of Life, an early version of what later became The Game of Life.

In 2004, he was posthumously inducted into the National Toy Hall of Fame along with George Ditomassi of Milton Bradley Company. Through the 20th century the company he founded in 1860, Milton Bradley Company dominated the production of American games, with titles like Candyland, Operation, and Battleship. The company is now a subsidiary of Pawtucket, Rhode Island-based Hasbro.

[edit] The Checkered Game of Life: The launch of the Milton Bradley Company

Milton Bradley's ventures into the production of board games began with a large failure in his lithograph business. When he attempted to print and sell copies of the presidential nominee Abraham Lincoln, Bradley initially met with great success. After they were released for sale, a customer contacted him calling it a fraud and demanding his money back because the picture was not an accurate representation of Lincoln, who had decided to grow his distinctive beard after Bradley's print was published. Suddenly, the prints were worthless, and Bradley burned those remaining in his possession.^[1] In search of a lucrative alternative project in which to employ his drafting skills, Bradley found inspiration from an imported board game given to him by a friend. Concluding that he could produce and market a similar game to American consumers, Milton Bradley released The Checkered Game of Life in the winter of 1860.

The game proved an instant success with the public. Bradley personally sold his first run of several hundred copies in one two-day period in New York; by 1861, consumers had bought over 45,000 copies. The Checkered Game of Life followed a structure similar to its American and British predecessors, with players spinning a teetotum to advance to corresponding squares. The squares each represented a social virtue or vice such as "influence" or "poverty", with the former earning a player points and the latter retarding his progress. However, even the most seemingly secure positions like "Fat Office" held the accompanying dangers of "Prison", "Ruin", and "Suicide".^[2] The player who first accumulated one hundred points won the game.

The original game board of The Checkered Game of Life.

While the structure of play used in The Checkered Game of Life differed little from previous board games, Bradley’s game embraced a radically different concept of success. Earlier children’s games, such as the popular Mansion of Happiness developed in Puritan Massachusetts, were concerned entirely with providing an attractive venue from which to promote moral virtue. But Bradley preferred to define success in secular business terms consistent with America’s emerging focus on “the causal relationship between character and wealth.” This approach, which depicted life as a quest for accomplishment in which personal virtues provided a means to an end, rather than a point of focus, complemented America’s burgeoning fascination with obtaining wealth in the years following the Civil War.

Final years[link]

Though The Checkered Game of Life and its several successive variations accounted for Bradley's financial success, board games did not constitute his primary focus in life. Once his pecuniary future had been secured, Bradley turned his attention to a series of progressive scientific and educational causes. Having met Edward Wiebe, an early American proponent of the kindergarten movement, in 1869, Bradley began to explore the ideas of the German romantic philosopher Friedrich Wilhelm August Fröbel. Fröbel challenged prevalent notions for educating children, which emphasized recitation, rote memorization, and the teaching of factual information from a child’s earliest schooling. Believing that these practices-–which attempted to instill an adult mentality in children-–ran contrary to both effective teaching and a child’s natural impulses, Fröbel suggested a pattern of education that focused on the child’s vantage point. Fröbel's theory stressed stimulation of aesthetic and sensory perception, kept lessons brief, presented them in simple terms suitable for a child’s consumption, and incorporated instinctual preferences for play and spontaneity. Froebel also used "gifts," which were objects to handle, such as geometric wooden blocks or balls and used in self-directed free play. He also used "occupations," which were materials to learn skills, later arts and crafts and also circles which included mother songs and finger plays. A year after Bradley met Wiebe, the Milton Bradley Company published Wiebe’s book “The Paradise of Childhood: A Manual for Self-Instruction in Freidrich Froebel’s Educational Principles, and a Practical Guide to Kinder-Gartners,” which describes a list of 20 Froebel gifts.

Enthralled with Fröbel's ideas, Bradley made distinct contributions to bringing them to prominence with the American public. Beginning in 1869, he published educational tracts and pamphlets on the virtues of Fröbel's kindergarten system. His company produced two magazines on the subject, Kindergarten News (later Kindergarten Review), and Work and Play. Though neither produced a profit, compelling Bradley’s business partners to withdraw their support, Bradley persevered, publishing the magazines until the end of his life. Bradley's lifelong friend, George Tapley, bought out his business partner's shares so Bradley could continue manufacturing his educational materials. By the 1890s, the Milton Bradley Company had introduced the first watercolour sets with standardized colours and education games such as Bradley's Word Builder and Bradley's Sentence Builder. The Milton Bradley Company was also the first to release crayon packages with standardized colors, forerunning the better-known Crayola Crayons and Artista Art Supplies of the Binney & Smith Company. Bradley's interest in art education also led him to produce a new color wheel for color instruction and publish four works about teaching colors.

Bradley married twice in his lifetime, first to Villona Eaton in 1860. They had no children together. He then married again to Ellen Thayer in 1869. Bradley and Ms. Thayer had two daughters together: Alice L. Bradley, born c. 1881, and Florence L. Bradley, born c. 1875. He died on May 30, 1911 in Springfield.

Books and inventions[link]

Books[link]

• Color in the Schoolroom, 1890
• Color in the Kindergarten, 1893
• Elementary Color, 1895
• Water Colors in the Schoolroom, 1900

Inventions[link]

• Bradley established a set of rules to play croquet in 1866.
• Bradley was one of the marketers of the zoetrope, a spinning slotted drum with pre-printed images to create the illusion of motion pictures.
• Though this was not definitely known as of November 2010, Bradley might have been awarded the patent for the one-arm paper cutter.

References[link]

• Bradley, Milton (1892). "The Color Question Again". Science (Moses King) 19 (477): 175–176. DOI:10.1126/science.ns-19.477.175-a. http://books.google.com/?id=yIMCAAAAYAAJ&pg=RA1-PA175. 
• Ing, Deborah (February 2000). "Bradley, Milton" (Reference Database - registration required). American National Biography Online. Oxford University Press. http://www.anb.org/articles/10/10-00177.html. Retrieved 2007-01-31. 
• Lepore, Jill. "The Meaning of Life: What Milton Bradley Started". The New Yorker, 21 May 2007, pp. 38–43..

External links[link]

• Milton Bradley at Find a Grave
• Milton Bradley: A Playful Legacy — slideshow by Life magazine
• The Vienna Maine Historical Society has purchased Bradley's birthplace and is in the process of relocating to the site, where it plans to open a small museum.