# What is Algebra? (Historically)

At the beginning of my second year, I thought it would be interesting to explore definitions of Algebra in historical texts: both how the author described Algebra and also what topics were included in Algebra. Unfortunately, I have not had time to do a more comprehensive search, but I found five British and American texts that were interesting and spanned about a century. (I chose to focus on these countries because the texts are readily available in digital form and also because I am limited in my language).

Please see below for additional details, but I list their definitions here:

• Hutton (1831) wrote, “Algebra is the science of investigation by means of symbols. It is sometimes also called Analysis; and is a general kind of arithmetic, or universal way of computation.”
• Smith (1848) defined Algebra, writing “By Algebra, which is a branch of mathematics, in which the quantities considered are represented by letters.” He continued, “Algebra dates a more recent origin than either Arithmetic or Geometry. It was not until Arithmetic had advanced to a considerable degree of perfection, and mathematicians had commenced to feel the necessity for abridging, as well as generalizing, its operations, that Algebra was introduced. In the early part of the 13th century, Leonardo, a merchant of Pisa, having made repeated visits to Arabia, returned to Italy with a knowledge of Algebra. A manuscript of his is quoted as far back as 1202.”
• Davies (1861) wrote that “Algebra is that branch of mathematics in which the quantities considered are represented by letters, and the operations to be performed upon them are indicated by signs. These letters and signs are called symbols.”
• Taylor (1889) wrote that “Algebra is the science of algebraic number and the equation. It differs from arithmetic (i.) in its number. The number of algebra has quality as well as arithmetical value. The double series of numbers in algebra gives a wider range to operations. Thus, in algebra to subtract a great quantity from a less is as natural as the reverse, while in arithmetic it is impossible. (ii.) In its symbols of number. Arithmetical symbols of number represent particular values; while in algebra any value may in general be attributed to the letters employed. Thus, arithmetic is confined to operations upon particular numbers, while algebra is adapted to the investigation of general principles. Again, in arithmetic all the different numbers which enter a problem are blended. (iii.) In its method of solving problems. In arithmetic we use identities, but very seldom equations; and a problem is solved by analyzing it. In algebra the characteristic instrument is the equation. To solve a problem, we translate its conditions into equations and solve those equations. The algebraic method renders easy the solution of many problems of which the arithmetical solution would be very difficult, or impossible.”
• Wentworth (1902) wrote that “In elementary Algebra we consider all quantities as expressed numerically in terms of some unit, and the symbols represent only the purely numerical parts of such quantities. In other words, the symbols denote what are called in Arithmetic abstract numbers.”

Charles Hutton

Charles Hutton had been a professor of Mathematics for the Royal Military Academy. You can see additional details about his life here: https://en.wikipedia.org/wiki/Charles_Hutton

In his “Course of Mathematics,” he included many Algebra topics, starting from basic operations, moving through roots and powers, to arithmetic and geometric progression, to some equations, and finishing with interest formulas.

Because his definition of Algebra depended on his definition of Mathematics, I include both here. He wrote, “Algebra is the science of investigation by means of symbols. It is sometimes also called Analysis; and is a general kind of arithmetic, or universal way of computation.”

Frances H. Smith, A.M.

Again, find additional details at https://en.wikipedia.org/wiki/Francis_Henney_Smith

Smith (1848) defined Algebra, writing “By Algebra, which is a branch of mathematics, in which the quantities considered are represented by letters.” He continued, “Algebra dates a more recent origin than either Arithmetic or Geometry. It was not until Arithmetic had advanced to a considerable degree of perfection, and mathematicians had commenced to feel the necessity for abridging, as well as generalizing, its operations, that Algebra was introduced. In the early part of the 13th century, Leonardo, a merchant of Pisa, having made repeated visits to Arabia, returned to Italy with a knowledge of Algebra. A manuscript of his is quoted as far back as 1202.”

Charles Davies, LL.D.

Find additional information about Charles Davies at the MAA website.

Davies (1861) wrote that “Algebra is that branch of mathematics in which the quantities considered are represented by letters, and the operations to be performed upon them are indicated by signs. These letters and signs are called symbols.”

J. M. Taylor, A.M.

James Morford Taylor served 57 years at Colgate University as a professor of Mathematics.

Taylor (1889) wrote that “Algebra is the science of algebraic number and the equation. It differs from arithmetic (i.) in its number. The number of algebra has quality as well as arithmetical value. The double series of numbers in algebra gives a wider range to operations. Thus, in algebra to subtract a great quantity from a less is as natural as the reverse, while in arithmetic it is impossible. (ii.) In its symbols of number. Arithmetical symbols of number represent particular values; while in algebra any value may in general be attributed to the letters employed. Thus, arithmetic is confined to operations upon particular numbers, while algebra is adapted to the investigation of general principles. Again, in arithmetic all the different numbers which enter a problem are blended. (iii.) In its method of solving problems. In arithmetic we use identities, but very seldom equations; and a problem is solved by analyzing it. In algebra the characteristic instrument is the equation. To solve a problem, we translate its conditions into equations and solve those equations. The algebraic method renders easy the solution of many problems of which the arithmetical solution would be very difficult, or impossible.”

George A. Wentworth

Wentworth (1902) wrote that “In elementary Algebra we consider all quantities as expressed numerically in terms of some unit, and the symbols represent only the purely numerical parts of such quantities. In other words, the symbols denote what are called in Arithmetic abstract numbers.”

# Why study the history of Mathematics Education? The case of multiplication.

I had the privilege to take a “History of the K-16 Mathematics Course” from Dr. Sharon Senk at Michigan State University. I loved the opportunities I had to dig into old mathematics textbooks – it’s amazing how many are available digitally through Google Scholar or other resources.

I argue that looking back at old textbooks helps me in thinking about mathematics education by broadening my perspective on what “traditional” means, how often the goals of textbook authors have changed, and giving me some additional support when I talk to parents and preservice teachers about why they need to learn “new-fangled” methods for understanding basic arithmetic operations.

I like to pull out some images I found in Swetz (1995), showing that multiple methods of multiplication (yes, even the “new-fangled” methods like the lattice method) were included in textbooks as early as the 13th century:

I like looking in old textbooks because they also help by allowing me to “rediscover” old methods or strategies that fell into disuse, and yet could be used to help students choose from additional methods, find methods that make more sense to them, and accept that mathematics isn’t a set of rigid procedures invented by a group of old math teachers somewhere, but that creativity can enter the mix – students can come up with their own strategies!

After looking at these methods, I also found this site that gives many other strategies for mulitplying:

https://threesixty360.wordpress.com/25-ways-to-multiply/

Reference

Swetz, F. (1995). To know and to teach: mathematical pedagogy from a historical context. Educational Studies in Mathematics, 29(1), 73-88.

# Solving Systems of Linear Equations: Changing Methods from 1820 – 2000

Looking through mathematics curriculum with a historical perspective has been interesting.  I am reminded that mathematics truly is evolving – the mathematical landscape is constantly changing.  When considering the topics that should be taught in algebra courses, or other mathematics courses as well, we should not throw topics away without careful consideration but we also should not keep topics only through loyalty to the past.  Unfortunately, it is difficult to understand fully the interdependence of topics and thus difficult to choose which topics are needed and which can be disposed of.

I began with a contemporary view of possible solution methods – graphical, tabular, matrix and elimination – and assumed that elimination methods have remained more or less stable (forever) and that matrix solutions (determinants) would also have been included even in early books.  I guessed that tabular and graphical methods would be used but not until mid-1900s.   Originally, I had especially been interested in the use of matrices in general in these textbooks, but it turns out that very few high school textbooks include use of matrices in general and then only as solution to systems of equations. I was not surprised to find that there are some consistent, basic elimination methods that have been taught for the past 200 years.  I was surprised to find that there had been additional methods that seem to have disappeared more recently.

For this project, I selected 82 algebra textbooks, intended for all levels of students, which were published in the United States between 1800 and 2000.  I attempted to create a list that was representative of typical algebra classroom texts. For textbooks published between 1800 and 1900, I based my selections on the catalog of algebra textbooks held by the Educational Research Library that was published in Svobodny (1985).  In the catalog were listed algebra textbooks including multiple reprints of some, as well as answer keys, exercise books and textbooks we would today consider as calculus. First, I went through the list to find textbooks that were intended as a treatment of algebra, including many that were ungraded (especially in the early 1800s) and many that included arithmetic, trigonometry or calculus as well as algebra.  I was able to find a digital copy of every title listed, even if occasionally the copy was an earlier printing or a reprint of the text.  I attempted to categorize each textbook according to the level of the intended audience (i.e., university, secondary or grammar) since the twentieth-century textbooks were all intended for high school students, however not every textbook was graded and not every author stated the intended audience.  Some textbooks were intended for any level from grammar-school to university students, but gave instructions that the younger student should begin by omitting certain sections or not attempting to understand the deeper mathematics on the first reading. In general, because systems of linear equations seemed to be acceptable for any level and because it was so difficult to separate the earlier textbooks into grade-level categories, I did not exclude the lower and higher levels. I feel this was worthwhile since I was then able to make comparisons between topics taught at different grade levels.

For textbooks published between 1900 and 2000, I used the list of “typical and other” algebra textbooks created and discussed in Donoghue (2003).  I was able to obtain a copy of each title on her list except John Swenson’s Integrated Mathematics with Special Applications to Algebra. Donoghue (2003) created this list based on a survey of algebra textbooks published in twentieth-century United States.  She included 2 or 3 textbooks for each twenty-year period with at least one textbook being an example of a “typical” high school algebra textbook and one being an example of an integrated or otherwise atypical algebra text.  In this case, Swenson’s text was considered atypical, so I felt that replacing it with a textbook published in that same timeframe would provide useful data, even if it was chosen for convenience (from Michigan State University Mathematics Library) rather than as being representative of a typical textbook.  I am worried that this made my sample less representative, but I felt it was already problematic. Because of this, I am not able to make strong claims about American algebra textbooks in general, but only the group of textbooks that I analyzed.  If I had the time to develop a better and more representative list of textbooks, I am still unsure whether that would be possible.

Once I had created this list of textbooks, and obtained digital or physical copies of each text on the list, I conducted a superficial survey of each.  In this superficial survey I recorded general data about the book, the author, the types of solution methods presented for systems of linear equations, and the types of systems of linear equations presented (number of unknowns, literal, numerical, application, etc.).  Regarding the author, I recorded the listed credentials (degree, position, previous textbooks published) as well as the author’s stated purpose for writing the textbook.

I did not analyze the credentials or stated purposes of the authors.  Without the numbers, I cannot state a certainty, but in general there seemed to be some differences between authors’ intents from the 1800s to the 1900s.  In the 1800s, the author generally referred to writing the textbook to follow a specific method: French (rigor or deeper mathematics); German (begins with the more advanced topic and then unpacks it); English (practical mathematics); subdivisions (breaking each topic into the smallest, most easily digested “chunks”); analytic or synthetic (rule-based); or inductive (“proceeds from the particular to the general” (Van Velzer & Slichter, 1890)) methods seemed to be the most common.  Even particular authors showed signs of changing. For example, Thomson and Quimby (1887) stated that they included many examples to “stimulate thought on the part of the student” but a few years later Thomson (1896) stated that he would not include so many examples that merely were “puzzles, calculated to waste the time and energy of the pupil.”

In contrast, in the 1900s, it seemed that authors often referred to the new educational theories, recommendations or standards, if they stated a purpose at all.   Many authors, whether in the 1800s or the 1900s, stated that students would learn to think by learning mathematics and others stated that in order to learn mathematics, students needed to learn to think. Authors in the 1800s seemed to be at all levels of education (A.B. to A.M. to Ph.D.) and placement (instructors at common schools, academies, normal schools and universities). If I had more time, I would analyze this in more detail to see if there are any trends over time related to these intents.

I saw a trend from the 1800s to 2000 of authors moving from presentation of algebraic methods (comparison, substitution, addition-subtraction, Bezout’s, division, etc.) to multiple representations (graphical, tabular, determinant, etc.). Of the methods employed, there were three or four methods that were taught for solving standard systems of equations before 1900: elimination through comparison, substitution, addition/subtraction and Bezout’s method (also called “method of undetermined multipliers” – rare in secondary algebra but relatively popular in college algebra).  Additional methods include: dividing one equation by the other, multiplying each side of an equation to the corresponding side of the other equation, and creating simpler equations using linear combinations of the equations in the set.

­

In Figure 1, the percentage of textbooks with a specific solution method for each twenty-year period is shown.  I provide brief descriptions ­­­­­of the less well-known methods below. It can be seen that the addition-subtraction and substitution methods seem to be stable since 1820.  On the other hand, the comparison method all but disappears in the mid-1900s, while the graphing and tabular methods gain popularity.  In the textbooks I looked at, determinants showed up briefly in the mid-1900s but then disappeared.  The increasing popularity of alternate representations, such as graphing, tables and determinants, is consistent with policy documents  throughout the twentieth century which called for multiple representations (MAA, 1923; NEA, 1894).

The “other methods” category includes a variety of solution methods: Bezout’s method (or undetermined multipliers), division methods, multiplication methods, methods using a general formula and methods using linear combination simplifications of the equations. These other methods, as well as the comparison, substitution and addition-subtraction methods are all algebraic in nature.  In the 1800s, almost every textbook included at least the comparison, substitution and addition-subtraction methods, usually starting with comparison.  In later books, the variety of methods used for elimination dwindled to substitution and addition/subtraction.  In the chart above, it can be seen that certain newer methods replaced the others – solutions using determinants, as well as graphical and numerical methods became standard. Only two books intended for secondary schools included determinant methods – one in 1897 and one in 1912.  Only one book intended for colleges (1898).  Four books intended for secondary schools included graphing methods: 1906, 1912 and 1920. No books intended for universities included graphing methods.

Comparison

The most popular method in the 1800s was solution by comparison, as illustrated by this image from Loomis (1870).  The method is to solve each equation for the same variable and then set them equal to each other and solve for the remaining variable.  I assume that the substitution method must generally result in fewer steps which is why this method fell out of favor in the popular algebra textbooks.

Linear Combinations

When given a name, some authors refer to this method as the method of “derived equations” (Collins, 1893):

This example is from Van Velzer and Slichter (1890).  The method is to add and subtract the given equations (or multiples thereof) in order to derive two new independent equations that are easier to deal with.  Once the new equations have been created, one of the more common methods of elimination is used to find the solution.

Bezout’s Method (Undetermined Multipliers)

The method of undetermined multipliers shows up relatively often in Algebra textbooks intended for college or secondary students in the 1800s. It is always referred to as the method of undetermined multipliers, but many authors also reference it as “Bezout’s method” or the “French Method.” This example from Collins (1893) shows the basic process.  First, the student would multiply one equation by an unknown value (represented by m). Next, add the transformed equation to the other equation.  Finally, choose a value for m so that one of the variables will be eliminated.

Division

This is a cute little method. First, manipulate the equations so all terms are on one side of the equals sign.  Then (to avoid fractions) multiply one equation so the coefficients are multiples of the initial coefficient of the other.  Divide the equation with the larger coefficient by the other. The result will be an expression with only one unknown. Set this equal to zero and solve for the unknown. Finally, use substitution to find the other unknown.

This example of the method of elimination by division is from Thomson and Quimby (1887).  This method is similar to the method of undetermined multipliers. In essence, we are subtracting a multiple of one equation from the other in a way that eliminates one variable, but the determination of which multiple to use is embedded in the standard division algorithm.

Multiplication

Only one textbook used the multiplication method, but I include it because it serves as an interesting example of the algebraic depth of some earlier textbooks. This example is from Van Velzer and Slichter (1888). I find this solution method interesting since it seems that to be only a mathematical exercise for the authors in equivalent systems of equations rather than an effective solution method. The authors point out that it will introduce additional solutions in certain cases (when each equation is equal to something other than a known quantity) so that the resulting system of equations is not necessarily equivalent. My interpretation of this is that it is intended to show students why certain methods are valid and why other methods are not, similar to teaching elementary students today about alternate base systems when learning base 10. This might help students think more deeply about the meaning of equivalent systems, which might help them later when solving systems of higher-degree equations.

This example shows the solution of a system of equations with two unknowns by multiplying each left-hand side by the other left-hand side and each right-hand side by the other right-hand side.  Because the author solved the first equation for x (instead of leaving both unknowns on the left-hand side), the solution introduced an extra solution as well.  This method could only be for curiosity’s sake since it does not always result in an equivalent set of equations and the process is very similar, but more complicated, than solving using the substitution method.

References

Collins, J. V. (1893). Text-book of algebra: Through quadratic equations. Chicago Albert, Scott & Company.

Dolciani, M. P., Berman, S. L., & Freilich, J. (1965). Algebra : Structure and Method: Houghton Mifflin.

Donoghue, E. F. (2003). Algebra and Geometry Textbooks in Twentieth-Century America. In G. M. A. Stanic & J. Kilpatrick (Eds.), A History of School Mathematics (Vol. 2, pp. 1629 – 1700). Reston, VA: National Council of Teachers of Mathematics.

Giffin, W. M. (1895). Grammar-school algebra: seventy-five suggestive lessons for beginners: Werner School Book Co.

Hutton, C., Adrain, R., & Gregory, O. (1831). A Course of Mathematics: For the Use of Academies as Well as Private Tuition : in Two Volumes: W.E. Dean.

Loomis, E. (1870). Elements of algebra: Harper & Bros.

MAA. (1923). The reorganization of mathematics in secondary education: a report of the National Committee on Mathematical Requirements under the auspices of the Mathematical Association of America, Inc.: The Mathematical Association of America, Inc.

Milne, W. J. (1920). High school algebra: American Book Co.

NEA. (1894). Report of the Committee of Ten on Secondary School Studies: With the Reports of the Conferences Arranged by the Committee: Published for the National Education Association, by the American Book Company.

Svobodny, D. (1985). Early American Textbooks, 1775-1900. A Catalog of the Titles Held by the Educational Research Library. Washington, D.C.: U.S. Department of Education.

Thomson, J. B. (1896). New practical algebra: adapted to the improved methods of instruction in schools, academies, and colleges with an appendix: Maynard.

Thomson, J. B., & Quimby, E. T. (1887). The collegiate algebra: adapted to colleges and universities. New York: Clark & Maynard, Publishers.

Van Velzer, C. A., & Slichter, C. S. (1888). A Course in Algebra: Being Course One in Mathematics in the University of Wisconsin. Madison, Wisconsin: Capital City Pub. Co., Printers.

Van Velzer, C. A., & Slichter, C. S. (1890). School algebra. Madison, Wisconsin: Tracy, Gibbs & Co.

Welchons, A. M., Krickenberger, W. R., & Pearson, H. R. (1976). Algebra, Book 1: Elementary Course. Lexington, MA: Ginn and Company.

Wells, W., & Hart, W. W. (1912). New High School Algebra. Boston: D. C. Heath & Co., Publishers.

# College Algebra: Synthetic Division

Author introduction: Both are mathematicians writing a new College Algebra textbook to address the changing needs of their students.  Both are concerned about keeping a rigorous textbook despite the changes being made.  Both wrote several mathematics textbooks, including Calculus.

James Taylor was a Mathematics professor at Colgate University from 1869; it was one of 24 American math textbooks – written by American and published in America. Ross Middlemiss taught at Washington University (in Missouri) from 1929 – 1969 and had a background in engineering mathematics.

Taylor (1892) and Middlemiss (1952)

I chose these College Algebra textbooks because the authors seemed to have similar motivations (see below) for writing, but in the 60 years between the texts, learning theories had changed and new recommendations (for secondary mathematics anyway) had been published:

• NEA. (1894).  Report of the Committee of Ten on Secondary School Studies: With the Reports of the Conferences Arranged by the Committee
• MAA. (1923). The reorganization of mathematics in secondary education: a report of the National Committee on Mathematical Requirements under the auspices of the Mathematical Association of America, Inc.
• NCTM. (1940).
Fifteenth Yearbook: The Place of Mathematics in Secondary Education.

Secondary education had opened to a more diverse population across that 60 years, and so a more diverse population was entering universities. The chart below illustrates that as the number of high school graduates in the United States increased, so did the number of enrolled college students.

Students with Diverse Backgrounds

Both authors had different definitions of “changing population”, but both wrote a new College Algebra to address similar issues.

Taylor (1892) mentioned the number of new sciences being added as fields of study and how mathematics must compensate by paring down content; that is, mathematics must be concise without losing rigor.

Middlemiss (1952) described the his new students: from the sciences, engineering, liberal arts, sociology, etc. and how he hoped to teach them all mathematical thinking and values of the mathematical discipline even if they wouldn’t need to remember the details of the content. He tried to appeal to them without losing rigor.

Both were careful to include most proofs, although both recommended the reader look at other references for complicated proofs.  Both asked student questions that are proof-like, or led to general conclusions.

His book seems written more to the student where Taylor seems written more to a teacher.  Middlemiss (following recommendations) used multiple representations, many illustrations, many application problems (connections to real world) as well as historical connections and very detailed explanations.

Middlemiss also gave a sample time table for the teacher and argued that it should be easily divided into 48 lessons of one hour each (in class) and two hours each (out of class).  Taylor argued that he carefully wrote his sections so that the teacher could change ordering without loss of students’ understanding.

Multiple Representations

The difference in use of multiple representations between the books is clear: Taylor (1892) provided one illustration in his textbook (support of his development of differentiation) while Middlemiss (1952) included multiple illustrations (triangles, rectangles, graphs) to support mathematical ideas. Middlemiss first included graphical representation with his introduction of Functions and Graphs, but then required students to provide graphical and/or tabular support for their arguments throughout the remainder of the text. Middlemiss also sometimes asked students to solve equations using two different methods, which seemed new compared to Taylor.

Synthetic Division, or Horner’s Method

William George Horner (1786 – 1837) was a British mathematician.  Both books refered to a numerical method for solving equations as “Horner’s method.” Taylor called synthetic division “Horner’s Method”.

Taylor presents synthetic division using an example of “standard” long division for a general cubic polynomial and then shows the process using the general coefficients.  I’ll admit – I’m not entirely sure I understand his explanation, but the process seems the same. He then gives three examples using numeric values before his student question section.

Middlemiss also began with the standard long division example. He then removed the x terms and showed only the coefficients. He then showed a simplified version, removing repeated terms and the coefficient “1” for x^2.

Finally, Middlemiss gave a step-by-step rule referring back to the original method and provided two two examples before moving on. As a curiosity, notice that both Taylor and Middlemiss had their divisor on the left. Taylor had divisor on left for synthetic and standard division, while Middlemiss had it on the right and explained only that we move it over for no other reason but convention. It seemed that the main difference was that Taylor was trying to be brief, while Middlemiss tried to be “friendly” to teachers or students who did not have a strong mathematical background.

# Evolution: Or, the Extraction of Roots.

I chose this topic to present in my “History of the K-16 Mathematics Curriculum” course, because I remember several teachers and professors mentioning that square root extraction had been taught as a standard topic up until a few years ago. Ever since, I’ve been curious how you can do so! I found that even though we do not need to find square roots manually anymore, a problem like this actually can connect ideas of length, area, trinomials, and number sense.

So, I looked back at two Arithmetic textbooks: Pike’s Arithmetic (1788) and Davies’ Arithmetic (1851). I chose these textbooks because Pike’s was rule-based and his stated goal was to present the rules, and use better illustrations and applications than had previously been available. Davies’ Arithmetic was published half a century later, after the “induction-synthetic” tug-of-war. Davies’ was still (or again?) rule-based, but the presentation method changed to: start with an overview, give definition, explain the propositions, give questions for students to answer mentally / orally, and to show connection of principles throughout each section. I felt that the biggest change between the two would be that Davies stressed he would explain the rules visually or concretely.

Over all, each textbook had some good and some bad: Davies did spend more time explaining to students what was happening in the rules, but Pike provided many more examples in a number of different fields. For example, above are three examples that require extraction of roots: arranging soldiers on a battle field, planting an orchard, and finding the diameter of a cistern pipe.

Pike’s rules for extraction of the square root, illustrating with the example 54756.

1. Rule 1 – Distinguish the given number into periods of two figure each, by putting a point over the place of units, another over the place of hundreds, and so on, which points shew the number of figures the root will consist of.
• Place a dot over the 6 (units place), 7 (hundreds place), and 5 (the root will be three figures):
2. Find the greatest square number in the first, or left hand period, place the root of it at the right hand of the given number, (after the manner of a quotient in division) for the first figure of the root, and the square number, under the period, and subtract it therefrom, and to the remainder bring down the next period for a dividend.
• The greatest square number in 5 (left-most digit) is 4. The root of 4 is 2, so place the digit 2 on the right of 54756 (similar to division).
• 5-4 = 1
• Bring down the next period for a dividend, means to bring down 47, so 147 is the new dividend.

• Place the double of the root, already found, on the left hand of the dividend for a divisor.
•  The double of 2 is 4, so we divide 147 by 40.  But – (in the next step) we find that 40 goes evenly into 147 three times, so the divisor will become 43.
• Seek how often the divisor is contained in the dividend, (except the right hand figure) and place the answer in the root for the second figure of it, and likewise on the right hand of the divisor: Multiply the divisor, with the figure last annexed, by the figure last placed in the root, and subtract the product from the dividend: To the remainder join the next period for a new dividend.
• So, right now the divisor is 40 and the dividend is 147. 40 divides into 147, 3 times.
• So “3” is the second digit of the root, and we add “3” to the divisor 40 – that is, the new divisor is 43
• Now, we multiply 43 * 3 = 129, and subtract 147 – 129 = 18
• We bring down the next period (56) and append that to 18, so the new dividend is 1856
• Double the figure already found in the root, for a new divisor, (or bring down your last divisor for a new one, doubling the right hand figure of it) and from these find the next figure in the root as last directed; and continue the operation in the same manner, till you have brought down all the periods.
• The last divisor was 43, and we need to double the “right hand figure” which is “3”, so we have 460 as the new divisor.
• 460 divides into 1856, 4 times.
• So “4” is the third digit of the root, and we add “4” to the divisor 460 – that is, the new divisor is 464.
• Now, we multiply 464 * 4 = 1856.
• So the square root of 54756 is 234.

That wasn’t too bad, right? It’s difficult to see why Pike’s Rules work, but it does make sense if you follow the reasoning using some algebra.

Notice above that Davies did not simply give his rules, but included a visual (square), some thinking questions (What is the greatest square of a single figure? In what places of figures will the square of the tens be found?), and an overall explanation: The square of two figures is equal to the square of the tens, plus twice the product of the tens by the units, plus the square of the units.  So, if a two-digit number is ab, where a is the first digit and b is the second (not multiplied but digits of a number), then (ab)^2 = (a0)^2 + 2*a0*b + (b)^2.

1. Point off the given number into periods of two figures each, counted from the right, by setting a dot over the place of units, another over the place of hundreds, and so on. [Similar to Pike’s so far.]
2. Find the greatest square in the first period on the left, and place its root on the right after the manner of a quotient in division. Subtract the square of the root from the first period, and to the remainder bring down the second period for a dividend.
3. Double the root already found and place it on the left for a divisor. Seek how many times the divisor is contained in the dividend, exclusive of the right-hand figure, and place the figure in the root and also at the right of the divisor.
4. Multiply the divisor, thus augmented by the last figure of the root, and subtract the product from the dividend, and to the remainder bring down the next period for a new dividend. But if the product should exceed the dividend, diminish the last figure of the root.
5. Double the whole root already found, for a new divisor, and continue the operation as before, until all the periods are brought down.

Davies illustrates the meaning of his rules by providing a visual of a perfect square, in this case, illustrating that the square root of 1296 is 36.

1. That is, first the student would find the largest square less than 12, which is 9, so start with 900 in the large square.
2. But the square root of 900 is just 30. What’s left? 1296 – 900 = 396.
3. From the visual, we can see that we’ll have two rectangles with one length 30 and the other is unknown, but the same value for each. So multiple 30*2 = 60. So how many times can 60 go evenly into 496? 6 times.
4. Multiply 60*6 = 360, so the two side rectangles have area 180, and all that is left is the 6*6 = 36.
5. So the square root of 1296 = 900 + 2*180 + 6^2 = 36^2.
Recall from above that, (36)^2 = (30)^2 + 2*30*6 + (6)^2 = 900 + 2*180 + 36.

Let’s see if Davies’ visual can help us make sense of Pike’s process with 54756. First of all, why put dots over the digits?

From Pike, we know the dots divide the number into “periods” of 56, 47, and 5 (moving from right to left). Another way to think of the periods is as the numbers 50000, 4700, and 56. The first number, 50000 gives us a way to find the area and side lengths of the largest square:

1. So, even though we do not need to find square roots manually anymore, a problem like this actually can connect ideas of length, area, trinomials, and number sense. Remembering that we are anticipating that the root of a five digit number must be three digits, say a represents the hundreds digit, b represents the tens digit, and c  represents the units digit. Then the root of 54756 is (a+b+c).
2. So we find a by asking: What is the largest perfect square less than 5? 4 is, so we have the area of the dark blue square is a^2 = 40000 = (200)^2, so a = 200.  Subtracting off the area of this square from the area of the whole square, we have 54,756-40000=14,756.
3. We’ll find b by asking: How many times does the double of a (400) go into 14,756?
4. So, we divide 14,756 by 400.
But, remembering that b represents the tens place of our final answer, it will be some digit followed by a zero. So, even though 400 goes into 14,756 evenly 36 times, we’ll round down to b = 30.  Now we find the size of the square consisting of a^2 + 2*a*bb^2 = 40000 + 12000 + 900 = 52,900. So 54,756 – 40,000 = 14,756.  And 14,756 – 12000 – 900 = 1856.
5. Finally, find c by asking: How many times does the double of (a + b), that is, 2*(200 + 30) = 460, go into 1856?  460 goes into 1856 evenly 4 times, so c = 4. And our final answer is 234.
Note that each time we have simply found the missing length of a rectangle, and found thus that:
(a + b + c)^2 = a^2 + 2*a*bb^2 + 2*(a + b)*cc ^2.  So, even though we do not need to find square roots manually anymore, a problem like this actually can connect ideas of length, area, trinomials, and number sense.