You missed the point.
It's not his boss that he needs to convince. He said his boss agrees.
It's the CEO that needs to be convinced. But the CEO has already said that the size of the IT department is adequate for the size of the company. As has been discussed in other threads, he's probably right as well when it comes to purely IT support related needs.
On the other hand, if the other departments have a need for specialized software development and has a proper business case for that software and the investement required to develop it, if he wants to convince his CEO that they need more developers for these projects, its the business owners that need to persuade the CEO.
At the end of the day, IT departments have a general budget for general IT costs. Which is seen as a cost center. And should be x% of the overall revenue of the company, or x% of the total number of employees that are supported.
If company revenue doesn't increase there's no capacity to increase the budget of a cost center that will not return easily identifiable profits.
Instead, the more effective argument, is to have the business become stakeholders in the projects they want with their own budgets to pay out of their budget (no longer is it a company cost, but rather a departmental investement) for the resources borrowed from the IT department to work on their systems.