This is an App designed to model the world’s first ever computer program published in 1843, by Ada Lovelace, daughter of Lord Byron. The text below explains what we are seeing.

The App and indeed Ada’s own program is modelled on the information and findings of her famous section of text called note G. Ada’s work there is essentially a table of 17 columns and 25 rows in which numbers are shown to be multiplied, divided, added or subtracted across the columns. The reading down the column corresponds to the progress of the calculation through time and to the literal movement of gears in the machine which was to perform these operations. The machine was Charles Babbage’s Analytical Engine. Charles Babbage was a good friend of Ada Lovelace. Built in part and fully documented in engineering drawings, it would only actually be fully reconstructed as a working model after the deaths of both of these visionaries. Nevertheless ongoing reconstruction of this important historical moment has helped to highlight the importance of the program too

The program even has the dubious distinction of containing the world’s first ever computer bug, an upside down fraction causing it to generate an incorrect string of results! These need to be fixed before the App gives the correct values. Thankfully, you just have to click on two tick boxes at the top of the page, correcting operations 4 and 24, to do this.

The App simulates the progression down the column of operations at the press of a key. As you click through, you can see the contents of the computer’s memory and the operations being performed. Remember Ada’s table only had 17 columns and that corresponds to 17 bytes of memory being used, yes that’s bytes not megabytes!

Check out Ada’s table of columns and rows with this reconstruction →

The purpose of Ada’s program, apart from being an illustration of what a computer program might actually look like to an 1843 reader, is to create a sequence of numbers. This is the Bernouilli sequence and mathematicians, including Bernouilli himself, noted they kept popping up when calculations were made for summing up long sequences in different areas of maths.

They became prized, not only for their usefulness in shortcutting laborious multiplication in these sequences, but also because they were notoriously difficult to calculate themselves. Once calculated, they went into the world’s mathematical tables and served as valuable basecamps for these long calculations. At the time of Ada in 1843, only 15 had been worked out. With modern computers the number is 100,000,000. Who knows how far Ada would have reached if Babbage had actually been able to build his machine and Ada had been able to run her program on it.

You will see the operations described as cards. The cards resembled those that were used in the modern Jacquard, textile looms of the day. Each card in Ada’s program would initiate its particular operation, one after another, cycling through the whole stack of 25. These cards could also be counted backwards in a single wadge if a particular condition were met, such as a counter not yet reaching zero. You can see this simulated in the App, where the cards 13 to 23, reappear a second time. In modern day terminology this is called a loop and Ada’s program used them. To calculate the Bernouilli numbers, many of these loops were put together, multiplying the computing power without any mental effort from the manual operator.

Finally, to summarise Ada’s note G, after 25 cards, one loop and 36 steps in all, the relatively early Bernouilli number, known as Ada’s B7*, pops out. It’s value is -1/30. Note G is just a step by step guide of the calculation of this particular Bernouilli number. But her method is applicable to all the Bernouilli numbers that follow.

Ada’s B7 is in fact the earliest Bernouilli number that uses the looping system of her program and all subsequent Bernouillis use more and more of these loops. The reason for this is that the Bernouillis themselves had to be calculated from all the previous Bernouilli numbers that had come before them and further, each of these numbers in turn had to be multipled by a chain of ever increasing length made up of fractions, essentially 8/2×7/3×6/4×5/5… until the cycle for that Bernouilli number had ended. Then it would be on to the next Bernouilli number and so on.

For orientation, we are placed early in the Bernoulli sequence here. In other words in 1843, of the 15 Bernouillis that had been found, Ada was cutting her teeth on a repetitive method that could calculate the 4th* Bernouilli from the previous 3. There would be 11 more Bernouillis to go after that, before she would have been breaking into the unknown value of the 16th Bernouilli number and then ever onwards.

*A note on the numbering. The App refers to Ada’s B7 and that is clearly at odds with the idea that it was the 4th number to be found. Elsewhere you will see a reference to the modern convention of describing it as the 8th number, this doubling due to a convention that alternate Bernouillis must be counted even though they always equal zero. These are all different labels, designated at different times for the same thing.