Is it possible to get old tweets through api?

Question

PHP

Hello. There is a website that has a twitter feed. The JS script accesses codebird php via ajax, requests tweets and inserts them into the page. I would like to receive tweets by hash tag, but old tweets will not be scraped. As I understood from here:
https://dev.twitter.com/rest/reference/get/search/...
there are several parameters:
since_id
max_id
until
The first two, as I understand it, are only needed for pagination. until didn't help. Is there any other way to get old tweets by hash tag?