Oh No! I am sorry to hear that you lost your iPhone. I do apologize there is not way to track your phone unless you had the Find My iPhone feature enabled. Do you have insurance on your device? If so, you can file a claim at www.phoneclaim.com I hope you find your phone.
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The app doesn't need to be installed on the phone you wish to track, the tracking portion is built into iOS
The app is used to find other iDevices registered and set up under your Apple ID, for example, if you had an iPad and an iPhone, and you lost your iPad, you would install the app on your iPhone and use it to find your iPad, likewise, say you lost the iPhone, you would install the app on your iPad and use it to find your iPhone, if you have another iDevice, you can go this route
If you don't have another iDevice under the same Apple ID, you can also go to www.icloud.com on any computer or device with a web browser, log in using your Apple ID and password, and select "Find my IPhone", it will then bring up a map with the GPS location of your iPhone
Also, using Find my iPhone, you can select to send a sound so you can listen for it's location, even if you have ring/notification sound settings turned down or off