A Young Man's Mathematics Lessons

Pages of calculations and problem-solving in the lesson book of Daniel Toll. Image from Glen-Sanders Papers, collection on microfilm in the Grems-Doolittle Library (originals at New-York Historical Society in New York City). 

This blog entry is written by Schenectady County Historical Society trustee John Gearing.

Among the manuscript treasures of the Grems-Doolittle Library lies an oft-overlooked gem: the Glen-Sanders Papers. Members of the Glen and Sanders families resided in Scotia's eponymous mansion for over 200 years. In the mid-twentieth century the estate was broken up and the remaining Glen-Sanders papers came into the possession of the New-York Historical Society. The Schenectady County Historical Society has a copy of the papers on eighteen reels of microfilm. The collection includes correspondence and notes (the earliest of which is dated 1674), account books, maps, wills, and genealogical records.

A careful reading of such papers can help us better understand what life was like in Schenectady in earlier times. While one could be excused for assuming that eighteenth century life was far simpler and much less sophisticated than it is today, a mathematics lesson book in the Glen-Sanders papers suggests otherwise. The book bears the name of Daniel Toll and is dated 1790. The mathematics taught were practical in nature and covered topics essential to every successful merchant.

The first exercise taught young Mr. Toll how to subtract the weight of a container, the “tare,” to determine the weight its contents. For example, flour was sold at the wholesale level in barrels. This sounds simple, but some goods were sold in units of “hundredweights” (or 112 pounds) and tare was sometimes set at a percentage of the whole rather than the actual weight of the container. Conversions were often necessary. Also covered was the calculation of “Brokage,” which Toll defined as “the percentage charge levied by those called Brokers who find customers and selling them the goods of other men whether strangers or natives.” A budding merchant needed to calculate “tret” as well. Tret was the amount (typically 4 pounds per hundredweight) allowed for the wastage of goods during shipment.

Page dealing with the subjects of tare and tret in the lesson book of Daniel Toll. Image from Glen-Sanders Papers, collection on microfilm in the Grems-Doolittle Library (originals at New-York Historical Society in New York City).

Both simple and compound interest were covered in Daniel Toll's subjects. He was taught how to compute interest when the percentage was not a whole number, and how to compute either the return, the term, the principal, or the percentage when the other three factors were given. Fractions, both “vulgar” and decimal, were covered, along with multiplication.

The lessons were taught using pounds, shillings, and pence. Instead of being a decimal system like today's dollar, this system was based on multiples of twelve. Twenty pennies made one shilling, and twelve shillings made a pound. Merchants' calculations required converting pounds to shillings and pence, and vice versa. Some problems required converting everything to pence, completing the calculation in pence, and then reconverting the answer to pounds, shillings and pence.

Complex computations were taught using a sort of algorithm. For example, to determine the present value of a amount due to be paid in the future, Toll wrote:

“Answer
1. As 12 months are to the rate percent
So is the time proposed to a fourth number

2. Add that fourth number to ₤100

3. As that sum is to the fourth number
So is the given sum to the rebate

4. Subtract the rebate from the given sum
and the remainder is the present worth.”

Although this “answer” may be mystifying to modern eyes, a careful perusal of the Toll's sample problem shows that four steps above were easily translated into arithmetical calculations by students of the day.

Assuming the dates (1790 and 1793) in the lesson book are accurate, Daniel Toll would have been between 14 and 17 years old when learning the practical mathematics shown in this lesson book. The difficulty and complexity of Toll's math curriculum seems to compare favorably to that of today's students of the same age, suggesting that Schenectadians 224 years ago were not all that much different, in some respects, than we are today. Assuming that Jonathan Pearson's information is correct in his Genealogies of the Descendants of the First Settlers of Schenectady, Daniel Toll, it seems, grew up to be a physician, and not a merchant after all.