More mathematical conundrums solved with mental arithmetic and poetry too

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Regular readers of Roamer’s page have by now become quite well acquainted with the Reverend Isaiah Steen, a Presbyterian minister, mathematician and author of a once-popular school text book called Steen’s Mental Arithmetic.

He published his widely-referenced book, followed by several additional print-runs, when he was a teacher in the Royal Belfast Academical Institution in 1842.

Frontispiece of the Reverend Steen's book

Frontispiece of the Reverend Steen's book

The book is a rich mine of information about all things mathematical, introducing formulas, equations, procedures and techniques for solving numerical problems “in the mind” the author explained “without the use of paper or slate, or anything else on which to perform the operation.”

Some of Rev Steen’s mathematical manipulations seem almost magical today, relying on brain-power rather than our ubiquitous, battery-powered calculators that effortlessly multiply, divide, add and subtract.

It’s probably this dependence on modern technology that baffled some folk when faced with the Rev Steen’s Table of Interest, a little mathematical chart that was shared on this page last week.

Based on Professor James Thomson’s calculations, another of Inst’s mathematics doyens, Rev Steen explained in his book that the Table of Interest “may be useful in finding the number of days from any day of one month to any day of any other month.”

“It’s very interesting,” said an Armagh-reader, adding: “But I don’t understand it!” – sentiments shared by others, and Roamer!

My apologies – I only included Rev Steen’s above-mentioned introduction to the chart last week and omitted his demonstration of how it works.

Rev Steen offered four examples to aid his readers, preceded by a general rule – “the table gives the days between any day of any month and the same day of any other month, which must be increased or diminished by the days in excess or defect.”

In other words – if you want to know the number of days between today, December 12, and July 12, you just need to look at the December column of the chart and move horizontally across to where the vertical July column intersects. There you’ll find 212 – hey presto, the number of days between now and the 12th!

I’ve checked it on my calculator and Rev Steen is absolutely right, but that’s from the same date this month to the same date in July.

What happens when the dates are different?

Rev Steen’s four examples solved that irritating inconsistency quite simply, thus:

The first example asked, “find the days from March till October 22.”

Rev Steen deftly demonstrated his technique.

“In the left hand column we find March, opposite which, in the column having October at the top, we find the number 214. This is the number of days between March 15 and October 15, to which add the number of days from the 15th to the 22nd (which is seven) and we have the entire number – 221 days.”

Example two required the number of days from April 27 until December 18.

Once again, Rev Steen showed how it’s done.

“Opposite April, at the left, and under December, at the top, we find 244. This is the number of days from April 18 till December 18, from which subtract nine, the days from the 18th till 27th, and we have remaining 235.”

This is all fine and dandy, but the dates that Rev Steen demonstrated were in the same year.

What happens when the dates are different, as well as falling in different years?

In his third example of how his Table of Interest works, he asks: “How many days are there from June 12, 1845, till August 28, 1846?”

“Here, the days include a year, together with the days from June 12 till August 28 of the same year,” Rev Steen explains.

And he applies some mental arithmetic to his chart.

From June 12 till August 28 is 61 days on the chart plus another 16, the difference between the dates, added together giving 77.

“To this add a year, 365,” explains Rev Steen, “and we have 442.”

It’s ingenious, and Rev Steen even predicted the discrepancy that comes with a Leap Year!

“When February of a Leap Year occurs,” he advised, “add one day to the number of days found on the chart.”

And he provides an example of the Leap Year by asking the question: “How many days will it be from January 10 till Septemper 23, of the year 1848, which will be a Leap Year?”

“From January 10 till September 10,” he explains, “the table gives 243, to which add 13, for the days from the 10th till the 23rd of the month, together with another day, for February in a leap year being in the question, and we have 257.”

Somehow Roamer finds all this very comforting, knowing that away back in the 1840s Rev Steen’s little chart did everything that a modern computer can do.

Well, almost everything!

When I checked that his calculations between today and July 12 were correct my computer also informed me that there are 18,316,800 seconds, or 305,280 minutes, or 5,088 hours between now and the Twelfth!

And there was another comforting thought amidst Rev Steen’s complicated calculations – a ditty from my school days which I still use today, regularly, and which no pocket calculator can match:

“Thirty days are in September,

April, June, and in November;

February has twenty-eight alone;

And all the rest have thirty-one.”