So this is both a general and specific question.
First what are your favourite databases to subscribe too? I've just about exhausted the freely available sources for some subjects. I hated using databases back in college but I'm willing to give them another shot. What are some that worth the effort?
Onto the specific question. I am looking for some way to access a series of articles that appeared in scientific American in the mid-1800s (1859-1866). I've looked in project Gutenberg, LOC.gov & the Internet Archive, & Google Books to see if the specific issues are in their collection, no dice. Scientific American wants $8 per article and only give the title of the article along with the date etc, I have no idea is an article is long or short or if it's one of the right ones until it is purchased. So is there a database with all the old Scientific American articles already in it?
Unfortunately, if Scientific American themselves are charging access per issue to their back catalogue, then it's unlikely that you'll be able to find an equally comprehensive digitisation of their back catalogue anywhere else (whilst the copyright may have expired, Scientific American's publishers are quite entitled to charge for access to their digitised records). Do you have a university library nearby? Most let members of the public join for a reasonable annual fee, and they may subscribe to the digital collection or to other collections that interest you. Failing that, your best bet would probably be paying the cost of an annual or monthly subscription (which will work out cheaper than paying for individual articles or editions) so you can browse the documents at your leisure.
In terms of general databases, JSTOR is probably the best way to go if you're looking for a comprehensive database at minimal individual cost (though it is pricey). Whilst you can't get the full institutional version as an individual, you can get JPASS for either a monthly or annual fee; downloads are limited, but you get unlimited access to their online reading function. Although again, some university and some public libraries will give members of the public institutional access if you go to them physically. You can see the full collection here, which is pretty comprehensive - it's one of the best and easiest ways for researchers not attached to an institution to access back catalogues of journal articles at minimal cost.
Are you near to a public university by any chance? If you go to a public university's library, almost all of them are open to the public and allow on-site visitors to access databases for free.
I did find Cornell has put several issues in your time range online for free, this is public domain material so they can do that, and also ProQuest has 1845-1908 in their collection, and that's a major library database. But I'm hoping the Cornell collection suits your needs for free!