I would probably trust using a credit card online more than over the phone. Amazon.com and the other online book retailers have privacy policies and it's all automated. No human involvement. Over the phone you have the chance of the person being untrustworthy and using your info. The chances are slim, but you never know. Online you know that there's no middle man.
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And there you have a perfect illustration of the Generation Chasm!
I'd rather take a chance on one person than put info into any internet vehicle, no matter how secure they say it is! Ha!
Or, should my paranoia becoming especially active I would ask total cost and send a written order and a return address that is an anonymous Postal Box or an office address enclosing either a Bank check or International Money Order in payment ...