No scholarly links or background here but a first hand account of someone that was around in the bad old days of tax before computerized submission.
You got a form, you filled it in. There were instruction booklets of various degrees of helpfulness & help lines you could call. If paperwork wasn't your thing, well this is the reason H&R Block was created.
I'm in my 50s and always did my own returns on paper until 2012. Now that it's too awkward to obtain paper forms and I sometimes have trouble navigating the government web sites for PDFs, I get my taxes done professionally.
A packet containing all the forms and instructions I was likely to need arrived in the mail in January or February based on my previous year's return. I read the instructions. If I needed any forms not included in my packet, I went to any public library. The libraries had racks of quite a variety of forms and instructions. They also had copies of a big book called Package X, which contained all the forms and their instructions. I xeroxed the pages I figured I would need.
The government also provided free guide booklets, as opposed to line-by-line instruction booklets, for complicated situations. These were listed on the back (or inside cover?) of the 1040 instruction booklet. I called an 800 number to have a couple guide booklets sent to me every few years. Or I could usually find those guide booklets at the library and read them there and xerox the crucial pages.
Bookstores also had a section of for-sale tax guide books from a few publishers. Some years I'd buy a book, but I can't remember the publisher. Maybe Kiplinger?
Disclosure: I took a full year of courses from H&R Block in the early 1980s and worked there a few years. Block had their own paper forms that said H&R Block on them. So, besides my own, I've prepared over 1000 returns (including a few fiduciary and corporate) on paper. Being able to work a 10-key adding machine by touch is crucial. As for checking for math errors, go through the papers and finished forms backwards, subtracting where you would normally add and adding where you would normally subtract. You should end up with 0. IIRC, you are off by 9 if you accidentally transposed a couple digits in your math. If you are off by any other value, you really messed up, maybe copying a number wrong in the first place. Fix the numbers, fix the math. Then the returns were packed in a suitcase and taken to Block's local checking office, where they were checked for logical errors and fishiness.
This probably violates some rules but I'm not really sure how to answer this as a historian. I'll assume for this discussion that you mean filing individual income taxes in the U.S.
Each year there was (and is) a standard form (in the case of federal income taxes, that's usually Form 1040). There was (and is) an instruction booklet that tells you how to fill in each line, when you need a supplemental form, etc.
(PDF warning)
Form 1040 in 1950. Here are the Instructions in 1950
Form 1040 in 1990. Here are the instructions for 1990
For reference, Form 1040 in 2015. Here are the instructions for 2015
Obviously taxes aren't always simple to prepare and accounting and tax professionals are necessary in many cases, but computers are not a requisite for figuring out tax liabilities, even if they make the process easier.
Incidentally, I do my taxes by hand every year, including this year. I have family members who prepare taxes professionally and they've only exclusively used software for less than a decade.